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Publications

  • Generalized permutahedra and optimal auctions (, and )
    SIAM Journal on Applied Algebra and Geometry, .
    Bibtex Abstract
    @article{JoswigKlimmSpitz2022,
    author = {Michael Joswig and Max Klimm and Sylvain Spitz},
    title = {Generalized permutahedra and optimal auctions},
    journal = {SIAM Journal on Applied Algebra and Geometry},
    year = {to appear},
    accepted = {30.09.2022},
    }
    We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.
  • Reduction of Potential-Based Flow Networks (, , and )
    Mathematics of Operations Research, .
    Bibtex Abstract
    @article{KlimmPfetschRaber+2022,
    author = {Max Klimm and Marc Pfetsch and Rico Raber and Martin Skutella},
    title = {Reduction of Potential-Based Flow Networks},
    journal = {Mathematics of Operations Research},
    year = {to appear},
    accepted = {29.10.2022},
    }
    We consider potential-based flow networks with terminal nodes at which flow can enter or leave the network and physical properties such as voltages or pressures are measured and controlled. We study conditions under which such a network can be reduced to a smaller, equivalent network with the same behavior at the terminal nodes. Potential-based flow networks are widely used to model infrastructure networks such as electricity, gas, or water networks. In contrast to Kron's reduction for electrical networks, we prove that, in general, potential-based flow networks with at least three terminals cannot be reduced to smaller networks whose size only depends on the number of terminals. On the other hand, we show that it is possible to represent a special class of potential-based flow networks by a complete graph on the terminals, and we establish a characterization of networks that can be reduced to a path network. Our results build on fundamental properties of effective resistances proved in this paper, including explicit formulae for their dependence on edge resistances of the network and their metric properties.
  • Optimal Impartial Correspondences (, and )
    WINE 2022 – Proc. 18th Conference on Web and Internet Economics.
    Bibtex Abstract
    @inproceedings{CembranoFischerKlimm2022,
    author = {Javier Cembrano and Felix A. Fischer and Max Klimm},
    booktitle = {WINE 2022 – Proc. 18th Conference on Web and Internet Economics},
    title = {Optimal Impartial Correspondences},
    year = {to appear},
    accepted = {07.09.2022},
    }
    We study mechanisms that select a subset of the vertex set of a directed graph in order to maximize the minimum indegree of any selected vertex, subject to an impartiality constraint that the selection of a particular vertex is independent of the outgoing edges of that vertex. For graphs with maximum outdegree $d$, we give a mechanism that selects at most $d+1$ vertices and only selects vertices whose indegree is at least the maximum indegree in the graph minus one. We then show that this is best possible in the sense that no impartial mechanism can only select vertices with maximum degree, even without any restriction on the number of selected vertices. We finally obtain the following trade-off between the maximum number of vertices selected and the minimum indegree of any selected vertex: when selecting at most $k$ of vertices out of $n$, it is possible to only select vertices whose indegree is at least the maximum indegree minus $\lfloor(n-2)/(k-1)\rfloor+1$.
  • Multidimensional Political Apportionment (, and )
    Proceedings of the National Academy of Sciences, 119(15):e2109305119, .
    DOI Preliminary version (EC 2021) Bibtex Abstract
    @article{CembranoCorreaVerdugo2022,
    author = {Javier Cembrano and Jose Correa and Victor Verdugo},
    doi = {10.1073/pnas.2109305119},
    journal = {Proceedings of the National Academy of Sciences},
    number = {15},
    pages = {e2109305119},
    title = {Multidimensional Political Apportionment},
    volume = {119},
    year = {2022},
    }
    Deciding how to allocate the seats of a deliberative assembly is one of the most fundamental problems in the political organization of societies and has been widely studied over two centuries already. The idea of proportionality is at the core of most approaches to tackle this problem, and this notion is captured by the divisor methods, such as the Jefferson/D'Hondt method. In a seminal work, Balinski and Demange extended the single-dimensional idea of divisor methods to the setting in which the seat allocation is simultaneously determined by two dimensions and proposed the so-called biproportional apportionment method. The method, currently used in several electoral systems, is, however, limited to two dimensions and the question of extending it is considered to be an important problem both theoretically and in practice. In this work we initiate the study of multidimensional proportional apportionment. We first formalize a notion of multidimensional proportionality that naturally extends that of Balinski and Demange. By means of analyzing an appropriate integer linear program we are able to prove that, in contrast to the two-dimensional case, the existence of multidimensional proportional apportionments is not guaranteed and deciding their existence is a computationally hard problem ($\mathsf{NP}$-complete). Interestingly, our main result asserts that it is possible to find approximate multidimensional proportional apportionments that deviate from the marginals by a small amount. The proof arises through the lens of discrepancy theory, mainly inspired by the celebrated Beck–Fiala theorem. We finally evaluate our approach by using the data from the recent 2021 Chilean Constitutional Convention election.
  • On the Robustness of Potential-Based Flow Networks (, , and )
    Mathematical Programming, .
    PDF DOI Bibtex Abstract
    @article{KlimmPfetschRaber+2022a,
    author = {Max Klimm and Marc E. Pfetsch and Rico Raber and Martin Skutella},
    doi = {10.1007/s10107-021-01760-w},
    journal = {Mathematical Programming},
    title = {On the Robustness of Potential-Based Flow Networks},
    year = {2022},
    }
    Potential-based flows provide a simple yet realistic mathematical model of transport in many real-world infrastructure networks such as, e.g., gas or water networks, where the flow along each edge depends on the difference of the potentials at its end nodes. We call a network topology robust if the maximal node potential needed to satisfy a set of demands never increases when demands are decreased. This notion of robustness is motivated by infrastructure networks where users first make reservations for certain demands that may be larger than the actual flows sent later on. In these networks, node potentials correspond to physical quantities such as pressures or hydraulic heads and must be guaranteed to lie within a fixed range, even if the actual amounts are smaller than the previously reserved demands. Our main results are a precise characterization of robust network topologies for the case of point-to-point demands via forbidden node-labeled graph minors, as well as an efficient algorithm for testing robustness.
  • Packing Under Convex Quadratic Constraints (, , and )
    Mathematical Programming, 192:361–386, .
    PDF DOI Preliminary version (IPCO 2020) Bibtex Abstract
    @article{KlimmPfetschRaber+2022b,
    author = {Max Klimm and Marc E. Pfetsch and Rico Raber and Martin Skutella},
    doi = {10.1007/s10107-021-01675-6},
    journal = {Mathematical Programming},
    pages = {361--386},
    title = {Packing Under Convex Quadratic Constraints},
    volume = {192},
    year = {2022},
    }
    We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are $\mathsf{APX}$-hard to approximate and present constant-factor approximation algorithms based upon two different algorithmic techniques: a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation, and a greedy strategy. We further show that a combination of these techniques can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of these algorithms for problem instances arising in the context of real-world gas transport networks.
  • Parametric Computation of Minimum Cost Flows with Piecewise Quadratic Costs ( and )
    Mathematics of Operations Research, 47(1):812–846, .
    PDF DOI Preliminary version (SODA 2019) Bibtex Abstract
    @article{KlimmWarode2021,
    author = {Max Klimm and Philipp Warode},
    doi = {10.1287/moor.2021.1151},
    journal = {Mathematics of Operations Research},
    number = {1},
    pages = {812--846},
    title = {Parametric Computation of Minimum Cost Flows with Piecewise Quadratic Costs},
    volume = {47},
    year = {2022},
    }
    We develop algorithms solving parametric flow problems with separable, continuous, piecewise quadratic, and strictly convex cost functions. The parameter to be considered is a common multiplier on the demand of all nodes. Our algorithms compute a family of flows that are each feasible for the respective demand and minimize the costs among the feasible flows for that demand. For single commodity networks with homogenous cost functions, our algorithm requires one matrix multiplication for the initialization, a rank 1 update for each nondegenerate step and the solution of a convex quadratic program for each degenerate step. For nonhomogeneous cost functions, the initialization requires the solution of a convex quadratic program instead. For multi-commodity networks, both the initialization and every step of the algorithm require the solution of a convex program. As each step is mirrored by a breakpoint in the output this yields output-polynomial algorithms in every case.
  • Equilibria in Multi-Class and Multi-Dimensional Atomic Congestion Games ( and )
    Mathematics of Operations Research, .
    PDF DOI Preliminary version (WINE 2014) Bibtex Abstract
    @article{KlimmSchuetz2022,
    author = {Max Klimm and Andreas Sch{\"u}tz},
    doi = {10.1287/moor.2021.1223},
    journal = {Mathematics of Operations Research},
    title = {Equilibria in Multi-Class and Multi-Dimensional Atomic Congestion Games},
    year = {2022},
    }
    This paper studies the existence of pure Nash equilibria in atomic congestion games with different user classes where the cost of each resource depends on the aggregated demand of each class. A set of cost functions is called consistent for this class if all games with cost functions from the set have a pure Nash equilibrium. We give a complete characterization of consistent sets of cost functions showing that the only consistent sets of cost functions are sets of certain affine functions and sets of certain exponential functions. This characterization is also extended to a larger class of games where each atomic player may control flow that belongs to different classes.
  • Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium (, , , , and )
    Optimization Methods and Software, .
    PDF DOI Bibtex Abstract
    @article{GruebelHuberHuembs+2022,
    author = {Julia Gr{\"u}bel and Olivier Huber and Lukas H{\"u}mbs and Max Klimm and Martin Schmidt and Alexandra Schwartz},
    journal = {Optimization Methods and Software},
    title = {Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium},
    year = {2022},
    accepted = {20.06.2022},
    doi = {10.1080/10556788.2022.2117358},
    }
    Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems.
  • PICOS: A Python interface to conic optimization solvers ( and )
    Journal of Open Source Software, 7(70):3915, .
    PDF DOI Bibtex Abstract
    @article{SagnolStahlberg2022,
    author = {Guillaume Sagnol and Maximilian Stahlberg},
    doi = {10.21105/joss.03915},
    journal = {Journal of Open Source Software},
    number = {70},
    pages = {3915},
    title = {{PICOS}: A {Python} interface to conic optimization solvers},
    volume = {7},
    year = {2022},
    }
    Many convex and mixed integer constrained optimization problems can be compiled into a canonical form and handed to some off-the-shelf optimization solver for numeric solution. However, the necessary reformulations can be technically challenging and a program written to use one solver cannot easily be made to use another. PICOS is a well-established Python meta-interface to many proprietary and open source optimization solvers that allows problems to be input in a natural algebraic form and that handles solver selection and the required transformations transparently. This enables users to focus more on the high level optimization model and its application and less on technicalities.
  • Competitive Algorithms for Symmetric Rendezvous on the Line (, , and )
    SODA 2022 – Proc. 33rd ACM-SIAM Symposium on Discrete Algorithms, pp. 329–347.
    PDF DOI Bibtex Abstract
    @inproceedings{KlimmSagnolSkutella+2022,
    author = {Max Klimm and Guillaume Sagnol and Martin Skutella and Khai Van Tran},
    booktitle = {SODA 2022 – Proc. 33rd ACM-SIAM Symposium on Discrete Algorithms},
    doi = {10.1137/1.9781611977073.16},
    pages = {329--347},
    title = {Competitive Algorithms for Symmetric Rendezvous on the Line},
    year = {2022},
    }
    In the Symmetric Rendezvous Search on the Line with Unknown Initial Distance, two identical agents are placed on the real line with their distance, the other's location, and their orientation unknown to them. Moving along the line at unit speed and executing the same randomized search strategy, the agents' goal is to meet up as early as possible. The expected meeting time obviously depends on the unknown initial distance and orientations. The quality of a randomized search strategy is thus measured by its competitive ratio, that is, the ratio of the expected meeting time and the earliest possible meeting time (half the initial distance). We present a class of successively refined randomized search strategies together with a rigorous mathematical analysis of their continuously improved competitive ratios. These strategies all rely on the basic idea of performing an infinite sequence of steps of geometrically increasing size in random directions, always returning to the agent's initial position before starting the next step. In addition, our more refined strategies use two novel ideas. First, remembering their past random choices, the agents randomly choose the direction of the next step in a Markov-chain-like manner. Second, choosing the next few random directions in advance, each agent may combine consecutive steps in the same direction into one longer step. As our main result, we show that this combination of looking into the past as well as into the future leads to a substantially improved competitive ratio of 13.93 compared to the previously best known bound of 24.85 (Ozsoyeller et al. 2013).
  • Impartial Selection with Additive Guarantees via Iterated Deletion (, , and )
    EC 2022 – Proc. 23rd ACM Conference on Economics and Computation, pp. 1104–1105.
    DOI Bibtex Abstract
    @inproceedings{CembranoFischerHannon+2022,
    author = {Javier Cembrano and Felix A. Fischer and David Hannon and Max Klimm},
    booktitle = {EC 2022 – Proc. 23rd ACM Conference on Economics and Computation},
    doi = {10.1145/3490486.3538294},
    pages = {1104--1105},
    title = {Impartial Selection with Additive Guarantees via Iterated Deletion},
    year = {2022},
    }
    Impartial selection is the selection of an individual from a group based on nominations by other members of the group, in such a way that individuals cannot influence their own chance of selection. We give a deterministic mechanism with an additive performance guarantee of $\mathcal{O}(n^{(1+\kappa)/2})$ in a setting with $n$ individuals where each individual casts $\mathcal{O}(n^\kappa)$ nominations, where $\kappa \in [0,1]$. For $\kappa =0$, i.e. when each individual casts at most a constant number of nominations, this bound is $\mathcal{O}(\sqrt{n})$. This matches the best-known guarantee for randomized mechanisms and a single nomination. For $\kappa=1$ the bound is $\mathcal{O}(n)$. This is trivial, as even a mechanism that never selects provides an additive guarantee of $n-1$. We show, however, that it is also best possible: for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.
  • Public Signals in Network Congestion Games (, , and )
    EC 2022 – Proc. 23rd ACM Conference on Economics and Computation, pp. 736.
    PDF DOI arxiv Bibtex Abstract
    @inproceedings{GriesbachHoeferKlimm+2022,
    arxiv = {2205.09823},
    author = {Svenja M. Griesbach and Martin Hoefer and Max Klimm and Tim Koglin},
    booktitle = {EC 2022 – Proc. 23rd ACM Conference on Economics and Computation},
    doi = {10.1145/3490486.3538349},
    pages = {736},
    title = {Public Signals in Network Congestion Games},
    year = {2022},
    }
    We consider a largely untapped potential for the improvement of traffic networks that is rooted in the inherent uncertainty of travel times. Travel times are subject to stochastic uncertainty resulting from various parameters such as weather condition, occurrences of road works, or traffic accidents. Large mobility services have an informational advantage over single network users as they are able to learn traffic conditions from data. A benevolent mobility service may use this informational advantage in order to steer the traffic equilibrium into a favorable direction. The resulting optimization problem is a task commonly referred to as signaling or Bayesian persuasion. Previous work has shown that the underlying signaling problem can be NP-hard to approximate within any non-trivial bounds, even for affine cost functions with stochastic offsets. In contrast, we show that in this case, the signaling problem is easy for many networks. We tightly characterize the class of single-commodity networks, in which full information revelation is always an optimal signaling strategy. Moreover, we construct a reduction from optimal signaling to computing an optimal collection of support vectors for the Wardrop equilibrium. For two states, this insight can be used to compute an optimal signaling scheme. The algorithm runs in polynomial time whenever the number of different supports resulting from any signal distribution is bounded to a polynomial in the input size. Using a cell decomposition technique, we extend the approach to a polynomial-time algorithm for multi-commodity parallel link networks with a constant number of commodities, even when we have a constant number of different states of nature.
  • Multi-Leader Congestion Games with an Adversary (, , , and )
    AAAI 2022 – Proc. 36th Conference on Artificial Intelligence, pp. 5068–5075.
    DOI Bibtex Abstract
    @inproceedings{HarksHenleKlimm+2022,
    author = {Tobias Harks and Mona Henle and Max Klimm and Jannik Matuschke and Anja Schedel},
    booktitle = {AAAI 2022 – Proc. 36th Conference on Artificial Intelligence},
    doi = {10.1609/aaai.v36i5.20439},
    pages = {5068--5075},
    tear = {2},
    title = {Multi-Leader Congestion Games with an Adversary},
    year = {2022},
    }
    We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads, causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria may fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a $K$-approximate equilibrium can always be guaranteed, where $K \approx 1.1974$ is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a $K$-approximate equilibrium. The factor $K$ is tight, meaning that there is an instance that does not admit an $\alpha$-approximate equilibrium for any $alpha < K$. Thus $\alpha = K$ is the smallest possible value of $\alpha$ such that the existence of an $\alpha$-approximate equilibrium can be guaranteed for any instance of the considered game. Secondly, we focus on approximate equilibria of a given fixed instance. We show how to compute efficiently a best approximate equilibrium, that is, with smallest possible $\alpha$ among all $\alpha$-approximate equilibria of the given instance.
  • Maximizing a Submodular Function with Bounded Curvature Under an Unknown Knapsack Constraint ( and )
    APPROX 2022 – Proc. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems.
    PDF DOI Bibtex Abstract
    @inproceedings{KlimmKnaack2022,
    author = {Max Klimm and Martin Knaack},
    title = {Maximizing a Submodular Function with Bounded Curvature Under an Unknown Knapsack Constraint},
    booktitle = {APPROX 2022 – Proc. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems},
    year = {2022},
    doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.49},
    accepted = {22.06.2022},
    }
    This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop an algorithm with a robustness factor that is decreasing in the curvature $c$ of the submodular function. For the extreme cases $c = 0$ corresponding to a modular objective, it matches a previously known and best possible robustness factor of $1/2$. For the other extreme case of $c = 1$ it yields a robustness factor of $\approx 0.35$ improving over the best previously known robustness factor of $\approx 0.06$.
  • Improved Approximation Algorithms for the Expanding Search Problem (, , and )
    Preprint, .
    PDF Bibtex Abstract
    @techreport{GriesbachHommelsheimKlimm+2022,
    author = {Svenja M. Griesbach and Felix Hommelsheim and Max Klimm and Kevin Schewior},
    title = {Improved Approximation Algorithms for the Expanding Search Problem},
    year = {2022},
    }
    A searcher faces a graph with edge costs and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge cost. The goal is to minimize the vertex-weighted sum of the exploration times of all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the case that all vertices have unit weight, we provide a $2e$-approximation. For the general case, we give a $(5e/2+\epsilon)$-approximation for any $\epsilon > 0$. Previously, for both cases only an $8$-approximation was known. Finally, we provide a PTAS for the case of a Euclidean graph.
  • Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling (, , and )
    Preprint, .
    PDF arxiv Bibtex Abstract
    @techreport{JaegerSagnolSchmidtgenanntWaldschmidt+2022,
    arxiv = {2211.02044},
    author = {Sven Jäger and Guillaume Sagnol and Daniel Schmidt genannt Waldschmidt and Philipp Warode},
    title = {Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling},
    year = {2022},
    }
    We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of $3$ for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any $b > 1$ a tight analysis for the natural $b$-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of $(1+3\sqrt{3})\approx 6.197$ for the deterministic and of $\approx 3.032$ for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is $2$-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees $<10$ for adaptions of the $b$-scaling strategy to online release dates and unweighted jobs on identical parallel machines.
  • Travelling on Graphs with Small Highway Dimension (, , and )
    Algorithmica, 83(5):1352–1370, .
    PDF DOI Preliminary version (WG 2019) Bibtex Abstract
    @article{DisserFeldmannKlimm+2021,
    author = {Yann Disser and Andreas E. Feldmann and Max Klimm and Jochen K{\"o}nemann},
    doi = {10.1007/s00453-020-00785-5},
    journal = {Algorithmica},
    number = {5},
    pages = {1352--1370},
    tier2 = {3},
    title = {Travelling on Graphs with Small Highway Dimension},
    volume = {83},
    year = {2021},
    }
    We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter roughly measures how many central nodes are visited by all shortest paths of a certain length. It has been shown that transportation networks, on which TSP and STP naturally occur for various applications in logistics, typically have a small highway dimension. While it was previously shown that these problems admit a quasi-polynomial time approximation scheme on graphs of constant highway dimension, we demonstrate that a significant improvement is possible in the special case when the highway dimension is 1. Specifically, we present a fully-polynomial time approximation scheme (FPTAS). We also prove that both TSP and STP are weakly $\mathsf{NP}$-hard for these restricted graphs.
  • Pure Nash Equilibria in Resource Graph Games (, and )
    Journal of Artificial Intelligence Research, 72:185–213, .
    PDF DOI Bibtex Abstract
    @article{HarksKlimmMatuschke2021,
    author = {Tobias Harks and Max Klimm and Jannik Matuschke},
    doi = {10.1613/jair.1.12668},
    journal = {Journal of Artificial Intelligence Research},
    pages = {185--213},
    title = {Pure {Nash} Equilibria in Resource Graph Games},
    volume = {72},
    year = {2021},
    }
    This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players' private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players' strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium.For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is $\mathsf{coNP}$-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is $\mathsf{\Sigma_2^p}$-complete.
  • Fractionally Subadditive Maximization under an Incremental Knapsack Constraint (, and )
    WAOA 2021 – Proc. 19th Workshop on Approximation and Online Algorithms.
    DOI Bibtex Abstract
    @inproceedings{DisserKlimmWeckbecker2021,
    author = {Yann Disser and Max Klimm and David Weckbecker},
    booktitle = {WAOA 2021 – Proc. 19th Workshop on Approximation and Online Algorithms},
    doi = {10.1007/978-3-030-92702-8_13},
    title = {Fractionally Subadditive Maximization under an Incremental Knapsack Constraint},
    year = {2021},
    editor = {J. Koenemann and B. Peis},
    }
    We consider the problem of maximizing a fractionally sub-additive function under a knapsack constraint that grows over time. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most $\max\{3.293\sqrt{M}, 2M\}$, under the assumption that the values of singleton sets are in the range $[1,M]$, and we give a lower bound of $\max\{2.449, M\}$ on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a tight bound of 2 for the incremental maximization of classical flows with unit capacities.
  • Proportional Apportionment: A Case Study From the Chilean Constitutional Convention (, , and )
    EAAMO 2021 - Proc. 1st ACM conference on Equity and Access in Algorithms, Mechanisms, and Optimization, pp. 1–9.
    DOI Bibtex Abstract
    @inproceedings{CembranoCorreaDiaz+2021,
    author = {Javier Cembrano and Jose Correa and Gonzalo Diaz and Victor Verdugo},
    booktitle = {EAAMO 2021 - Proc. 1st ACM conference on Equity and Access in Algorithms, Mechanisms, and Optimization},
    doi = {10.1145/3465416.3483295},
    pages = {1--9},
    title = {Proportional Apportionment: A Case Study From the Chilean Constitutional Convention},
    year = {2021},
    }
    How to elect the representatives in legislative bodies is a question that every modern democracy has to answer. This design task has to consider various elements so as to fulfill the citizens' expectations and contribute to maintaining a healthy democracy. In this work, we present five proposals of electoral methods based on proportional representation that extend the notion of proportionality to several dimensions: For instance, simultaneous proportionality in political, geographical, and gender representation. We also consider including two additional desirable properties for electoral methods: A minimum threshold to obtain political representation, and the incorporation of plurality voting, guaranteeing the election of the highest voted candidate in each district. We use the Chilean Constitutional Convention election (May 15–16, 2021) results as a testing ground, and compare the apportionment obtained under each method according to four criteria: Proportionality, representativeness, robustness, and voting power. We conclude that it is feasible to design and implement an electoral method satisfying all mentioned properties. Our findings may be useful in assessing electoral designs in other contexts as well.
  • Multidimensional Apportionment through Discrepancy Theory (, and )
    EC 2021 – Proc. 22nd ACM Conference on Economics and Computation, pp. 287–288.
    DOI Extended version (PNAS 2022) Bibtex
    @inproceedings{CembranoCorreaVerdugo2021,
    author = {Javier Cembrano and Jose Correa and Victor Verdugo},
    booktitle = {EC 2021 -- Proc. 22nd ACM Conference on Economics and Computation},
    doi = {10.1145/3465456.3467596},
    pages = {287--288},
    title = {Multidimensional Apportionment through Discrepancy Theory},
    year = {2021},
    }
  • Approximate Parametric Computation of Minimum-Cost Flows with Convex Costs (, and )
    Preprint, .
    PDF arxiv Bibtex Abstract
    @techreport{JoachimsKlimmWarode2021,
    arxiv = {2203.13146},
    author = {Per Joachims and Max Klimm and Philipp Warode},
    title = {Approximate Parametric Computation of Minimum-Cost Flows with Convex Costs},
    year = {2021},
    }
    This paper studies a variant of the minimum-cost flow problem in a graph with convex cost functions, where the demands at the vertices are functions depending on a one-dimensional parameter $\lambda$. We devise two algorithmic approaches for the approximate computation of parametric solutions for this problem. The first approach transforms an instance of the parametric problem into an instance with piecewise quadratic cost functions by interpolating the marginal cost functions. The new instance can be solved exactly with an algorithm we developed in prior work. In the second approach, we compute a fixed number of non-parametric solutions and interpolate the resulting flows yielding an approximate solution for the original, parametric problem. For both methods we formulate explicit bounds on the step sizes used in the respective interpolations that guarantee relative and absolute error margins. Finally, we test our approaches on real-world traffic and gas instances in an empirical study.
  • Hiring Secretaries over Time: The Benefit of Concurrent Employment (, , , , , , and )
    Mathematics of Operations Research, 45(1):323–352, .
    PDF DOI Bibtex Abstract
    @article{DisserFearnleyGairing+2020,
    author = {Yann Disser and John Fearnley and Martin Gairing and Oliver G{\"o}bel and Max Klimm and Daniel Schmand and Alexander Skopalik and Andreas T{\"o}nnis},
    doi = {10.1287/moor.2019.0993},
    journal = {Mathematics of Operations Research},
    number = {1},
    pages = {323--352},
    tier2 = {2},
    title = {Hiring Secretaries over Time: The Benefit of Concurrent Employment},
    volume = {45},
    year = {2020},
    }
    We consider a stochastic online problem where n applicants arrive over time, one per time step. Upon the arrival of each applicant, their cost per time step is revealed, and we have to fix the duration of employment, starting immediately. This decision is irrevocable; that is, we can neither extend a contract nor dismiss a candidate once hired. In every time step, at least one candidate needs to be under contract, and our goal is to minimize the total hiring cost, which is the sum of the applicants' costs multiplied with their respective employment durations. We provide a competitive online algorithm for the case that the applicants' costs are drawn independently from a known distribution. Specifically, the algorithm achieves a competitive ratio of 2.965 for the case of uniform distributions. For this case, we give an analytical lower bound of 2 and a computational lower bound of 2.148. We then adapt our algorithm to stay competitive even in settings with one or more of the following restrictions: (i) at most two applicants can be hired concurrently; (ii) the distribution of the applicants' costs is unknown; (iii) the total number n of time steps is unknown. On the other hand, we show that concurrent employment is a necessary feature of competitive algorithms by proving that no algorithm has a competitive ratio better than $\Omega(n/ \log n)$ if concurrent employment is forbidden.
  • Broadcasting a File in a Communication Network (, and )
    Journal of Scheduling, 23(2):211–232, .
    PDF DOI Preliminary version (ISAAC 2011) Bibtex Abstract
    @article{GoetzmannHarksKlimm2020,
    author = {Kai{-}Simon Goetzmann and Tobias Harks and Max Klimm},
    doi = {10.1007/s10951-020-00643-w},
    journal = {Journal of Scheduling},
    number = {2},
    pages = {211--232},
    title = {Broadcasting a File in a Communication Network},
    volume = {23},
    year = {2020},
    }
    We study the problem of distributing a file, initially located at a server, among a set of $n$ nodes. The file is divided into $m \geq 1$ equally sized packets. After downloading a packet, nodes can upload it to other nodes, possibly to multiple nodes in parallel. Each node, however, may receive each packet from a single source node only. The upload and download rates between nodes are constrained by node- and server-specific upload and download capacities. The objective is to minimize the makespan. This problem has been proposed and analyzed first by Mundinger et al. (J. Sched. 11:105–120, 2008) under the assumption that uploads obey the fair sharing principle, that is, concurrent upload rates from a common source are equal at any point in time. Under this assumption, the authors devised an optimal polynomial time algorithm for the case where the upload capacity of the server and the nodes' upload and download capacities are all equal. In this work, we drop the fair sharing assumption and derive an exact polynomial time algorithm for the case when upload and download capacities per node and among nodes are equal. We further show that the problem becomes strongly $\mathsf{NP}$-hard for equal upload and download capacities per node that may differ among nodes, even for a single packet. For this case, we devise a polynomial time $(1+ 2\sqrt{2})$-approximation algorithm. Finally, we devise two polynomial time algorithms with approximation guarantees of $5$ and $2 + \lceil \log_2 \lceil n/m \rceil \rceil /m$, respectively, for the general case of $m$ packets.
  • Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians ( and )
    SODA 2020 – Proc. 31st ACM-SIAM Symposium on Discrete Algorithms, pp. 2728–2747.
    PDF DOI Bibtex Abstract
    @inproceedings{KlimmWarode2020,
    author = {Max Klimm and Philipp Warode},
    booktitle = {SODA 2020 – Proc. 31st ACM-SIAM Symposium on Discrete Algorithms},
    doi = {10.1137/1.9781611975994.166},
    pages = {2728--2747},
    title = {Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block {Laplacians}},
    year = {2020},
    }
    We settle the complexity of computing an equilibrium in atomic splittable congestion games with player-specific affine cost functions $l_{e,i}(x) = a_{e,i}x + b_{e,i}$ showing that it is $\mathsf{PPAD}$-complete. To prove that the problem is contained in $\mathsf{PPAD}$, we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix. This leads to a path following formulation where states correspond to supports that are feasible for some demands and neighboring supports are feasible for increased or decreased flow demands. A closer investigation of the block Laplacian system allows to orient the states giving rise to unique predecessor and successor states thus putting the problem into $\mathsf{PPAD}$. For the $\mathsf{PPAD}$-hardness, we reduce from computing an approximate equilibrium of a bimatrix win-lose game. As a byproduct of our reduction we further show that computing a multi-class Wardrop equilibrium with class-dependent affine cost functions is $\mathsf{PPAD}$-complete as well. As a byproduct of our $\mathsf{PPAD}$-completeness proof, we obtain an algorithm that computes all equilibria parametrized by the players' flow demands. For player-specific costs, this computation may require several increases and decreases of the demands leading to an algorithm that runs in polynomial space but exponential time. For player-independent costs only demand increases are necessary. If the coefficients be,i are in general position, this yields an algorithm computing all equilibria as a function of the flow demand running in time polynomial in the output.
  • Packing Under Convex Quadratic Constraints (, , and )
    IPCO 2020 – Proc. 21st Conference on Integer Programming and Combinatorial Optimization, pp. 266–279.
    DOI Extended version (Math. Program. 2022) Bibtex Abstract
    @inproceedings{KlimmPfetschRaber+2020,
    author = {Max Klimm and Marc E. Pfetsch and Rico Raber and Martin Skutella},
    booktitle = {IPCO 2020 – Proc. 21st Conference on Integer Programming and Combinatorial Optimization},
    doi = {10.1007/978-3-030-45771-6\_21},
    pages = {266--279},
    title = {Packing Under Convex Quadratic Constraints},
    year = {2020},
    }
    We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are $\mathsf{APX}$-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding method leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three algorithms for problem instances arising in the context of real-world gas transport networks.
  • Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages (, , , , and )
    SoCG 2020 – Proc. 36th International Symposium on Computational Geometry, pp. 16:1–16:17.
    DOI Bibtex Abstract
    @inproceedings{BekosDaLozzoGriesbach+2020,
    address = {Dagstuhl, Germany},
    annote = {Keywords: Book embeddings, Book thickness, Nonplanar graphs, Planar skeleton},
    author = {Michael A. Bekos and Giordano Da Lozzo and Svenja M. Griesbach and Martin Gronemann and Fabrizio Montecchiani and Chrysanthi Raftopoulou},
    booktitle = {SoCG 2020 – Proc. 36th International Symposium on Computational Geometry},
    doi = {10.4230/LIPIcs.SoCG.2020.16},
    editor = {Sergio Cabello and Danny Z. Chen},
    pages = {16:1--16:17},
    title = {Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages},
    urn = {urn:nbn:de:0030-drops-121749},
    volume = {164},
    year = {2020},
    }
    An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs.
  • Nobel, Milgrom und Wilson ()
    Chapter in Mitteilungen der Deutschen Mathematiker-Vereinigung, .
    DOI Bibtex Abstract
    @incollection{Klimm2020,
    author = {Max Klimm},
    booktitle = {Mitteilungen der Deutschen Mathematiker-Vereinigung},
    doi = {10.1515/dmvm-2020-0062},
    number = {4},
    pages = {193--200},
    title = {Nobel, Milgrom und Wilson},
    volume = {28},
    year = {2020},
    }
    Paul Milgrom und Robert Wilson erhielten den Nobel-Gedächtnispreis für Wirtschaftswissenschaften für ihre Beiträge zur Auktionstheorie und die Entwicklung neuer Auktionsformate. Wir geben hier einen Überblick über die mathematische Analyse von Auktionen und den Einfluss von Milgrom und Wilson auf diese Theorien. Insbesondere werden wir der Frage nachgehen, nach welchen Prinzipien Auktionen gestaltet sein sollten, damit sie einen möglichst hohen Erlös erzielen.
  • SAGT 2020 – Proc. 13th International Symposium on Algorithmic Game Theory (Tobias Harks, Max Klimm, eds.), Springer, .
    DOI Bibtex
    @proceedings{HarksKlimm2020,
    doi = {10.1007/978-3-030-57980-7},
    editor = {Tobias Harks and Max Klimm},
    publisher = {Springer},
    series = {LNCS},
    title = {SAGT 2020 – Proc. 13th International Symposium on Algorithmic Game Theory},
    volume = {12283},
    year = {2020},
    }
  • Tight Bounds for Undirected Graph Exploration with Pebbles and Multiple Agents (, and )
    Journal of the ACM, 66(6):40:1–40:41, .
    PDF DOI Preliminary version (SODA 2016) Bibtex Abstract
    @article{DisserHackfeldKlimm2019,
    author = {Yann Disser and Jan Hackfeld and Max Klimm},
    doi = {10.1145/3356883},
    journal = {Journal of the ACM},
    number = {6},
    pages = {40:1--40:41},
    tier2 = {1},
    title = {Tight Bounds for Undirected Graph Exploration with Pebbles and Multiple Agents},
    volume = {66},
    year = {2019},
    }
    We study the problem of deterministically exploring an undirected and initially unknown graph with $n$ vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles $\Theta(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most $n$ vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory $\Theta(\log \log n)$ pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as $\Theta(\log \log n)$ agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any $n$-vertex graph using $\mathcal{O}(\log \log n)$ pebbles, even when $n$ is not known a priori. The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that $\mathcal{O}(\log \log n)$ agents with constant memory can explore any $n$-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already $\Omega(\log \log n)$ when we allow each agent to have at most $\mathcal{O}((\log n)^{1-ε})$ bits of memory for any $\epsilon > 0$. Our argument also implies that a single agent with sublogarithmic memory needs $\Theta(\log \log n)$ pebbles to explore any $n$-vertex graph.
  • Distance-Preserving Graph Contractions (, , , , and )
    SIAM Journal on Discrete Mathematics, 33(3):1607–1636, .
    DOI Preliminary version (ITCS 2018) Bibtex Abstract
    @article{BernsteinDaeubelDisser+2019,
    author = {Aaron Bernstein and Karl D{\"a}ubel and Yann Disser and Max Klimm and Torsten M{\"u}tze and Frieder Smolny},
    doi = {10.1137/18M1169382},
    journal = {SIAM Journal on Discrete Mathematics},
    number = {3},
    pages = {1607--1636},
    tier2 = {3},
    title = {Distance-Preserving Graph Contractions},
    volume = {33},
    year = {2019},
    }
    Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into supervertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding supervertices remain at distance at least $\varphi(d)$ in the contracted graph, where $\varphi$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $\varphi(x) = x/\alpha + \beta$, where $\alpha \geq 1$ and $\beta \geq 0$ are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (nonoptimal) contractions despite our hardness results.
  • Greedy Metric Minimum Online Matchings with Random Arrivals ( and )
    Operations Research Letters, 47(2):88–91, .
    PDF DOI Bibtex Abstract
    @article{GairingKlimm2019,
    author = {Martin Gairing and Max Klimm},
    doi = {10.1016/j.orl.2019.01.002},
    journal = {Operations Research Letters},
    number = {2},
    pages = {88--91},
    title = {Greedy Metric Minimum Online Matchings with Random Arrivals},
    volume = {47},
    year = {2019},
    }
    We consider online metric minimum bipartite matching problems with random arrival order and show that the greedy algorithm assigning each request to the nearest unmatched server is $n$-competitive, where $n$ is the number of requests. This result is complemented by a lower bound exhibiting that the greedy algorithm has a competitive ratio of at least $n^{(\ln 3 - \ln 2)/\ln 4} \approx n^{0.292}$, even when the underlying metric space is the real line.
  • A more fine-grained complexity analysis of finding the most vital edges for undirected shortest paths (, , , and )
    Networks, 73(1):23–37, .
    DOI Bibtex Abstract
    @article{BazganFluschnikNichterlein+2019,
    author = {Cristina Bazgan and Till Fluschnik and Andr{\'e} Nichterlein and Rolf Niedermeier and Maximilian Stahlberg},
    doi = {10.1002/net.21832},
    journal = {Networks},
    number = {1},
    pages = {23--37},
    title = {A more fine-grained complexity analysis of finding the most vital edges for undirected shortest paths},
    volume = {73},
    year = {2019},
    }
    We study the $\mathsf{NP}$-hard shortest path most vital edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices s and t, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest $st$-path as much as possible. This scenario has been studied from the viewpoint of parameterized complexity and approximation algorithms. We contribute to this line of research by providing refined computational tractability as well as hardness results. We achieve this by a systematic investigation of various problem-specific parameters and their influence on the computational complexity. Charting the border between tractability and intractability, we also identify numerous challenges for future research.
  • Computing all Wardrop Equilibria parametrized by the Flow Demand ( and )
    SODA 2019 – Proc. 30th ACM-SIAM Symposium on Discrete Algorithms, pp. 917–934.
    PDF DOI Extended version (MOR 2021) Bibtex Abstract
    @inproceedings{KlimmWarode2019,
    author = {Max Klimm and Philipp Warode},
    booktitle = {SODA 2019 – Proc. 30th ACM-SIAM Symposium on Discrete Algorithms},
    doi = {10.1137/1.9781611975482.56},
    pages = {917--934},
    title = {Computing all {Wardrop} Equilibria parametrized by the Flow Demand},
    year = {2019},
    }
    We develop an algorithm that computes for a given undirected or directed network with flow-dependent piece-wise linear edge cost functions all Wardrop equilibria as a function of the flow demand. Our algorithm is based on Katzenelson's homotopy method for electrical networks. The algorithm uses a bijection between vertex potentials and flow excess vectors that is piecewise linear in the potential space and where each linear segment can be interpreted as an augmenting flow in a residual network. The algorithm iteratively increases the excess of one or more vertex pairs until the bijection reaches a point of non-differentiability. Then, the next linear region is chosen in a simplex-like pivot step and the algorithm proceeds. We first show that this algorithm correctly computes all Wardrop equilibria in undirected single-commodity networks along the chosen path of excess vectors. We then adapt our algorithm to also work for discontinuous cost functions which allows to model directed edges and/or edge capacities. Our algorithm is output-polynomial in non-degenerate instances where the solution curve never hits a point where the cost function of more than one edge becomes non-differentiable. For degenerate instances we still obtain an output-polynomial algorithm computing the linear segments of the bijection by a convex program. The latter technique also allows to handle multiple commodities.
  • Travelling on Graphs with Small Highway Dimension (, , and )
    WG 2019 – Proc. 45th Workshop on Graph-Theoretic Concepts in Computer Science, pp. 175–189.
    DOI Extended version (Algorithmica 2021) Bibtex Abstract
    @inproceedings{DisserFeldmannKlimm+2019,
    author = {Yann Disser and Andreas Emil Feldmann and Max Klimm and Jochen K{\"o}nemann},
    booktitle = {WG 2019 – Proc. 45th Workshop on Graph-Theoretic Concepts in Computer Science},
    doi = {10.1007/978-3-030-30786-8\_14},
    editor = {Ignasi Sau and Dimitrios M. Thilikos},
    pages = {175--189},
    series = {LNCS},
    title = {Travelling on Graphs with Small Highway Dimension},
    volume = {11789},
    year = {2019},
    }
    We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly $|mathsf{NP}$-hard for these restricted graphs. For TSP we show $\mathsf{NP}$-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015].
  • The Online Best Reply Algorithm for Resource Allocation Problems (, and )
    SAGT 2019 – Proc. 12th International Symposium on Algorithmic Game Theory, pp. 200–215.
    DOI Bibtex Abstract
    @inproceedings{KlimmSchmandToennis2019,
    author = {Max Klimm and Daniel Schmand and Andreas T{\"o}nnis},
    booktitle = {SAGT 2019 – Proc. 12th International Symposium on Algorithmic Game Theory},
    doi = {10.1007/978-3-030-30473-7\_14},
    editor = {Dimitris Fotakis and Evangelos Markakis},
    pages = {200--215},
    series = {LNCS},
    title = {The Online Best Reply Algorithm for Resource Allocation Problems},
    volume = {11801},
    year = {2019},
    }
    We study the performance of a best reply algorithm for online resource allocation problems with a diseconomy of scale. In an online resource allocation problem, we are given a set of resources and a set of requests that arrive in an online manner. Each request consists of a set of feasible allocations and an allocation is a set of resources. The total cost of an allocation vector is given by the sum of the resources' costs, where each resource's cost depends on the total load on the resource under the allocation vector. We analyze the natural online procedure where each request is allocated greedily to a feasible set of resources that minimizes the individual cost of that particular request. In the literature, this algorithm is also known as a one-round walk in congestion games starting from the empty state. For unweighted resource allocation problems with polynomial cost functions with maximum degree $d$, upper bounds on the competitive ratio of this greedy algorithm were known only for the special cases $d \in \{1,2,3\}$. In this paper, we show a general upper bound on the competitive ratio of $d(d/W(\frac{1.2d-1}{d+1}))^{d+1}$ for the unweighted case where $W$ denotes the Lambert-$W$ function on $\mathbb{R}_{\geq 0}$. For the weighted case, we show that the competitive ratio of the greedy algorithm is bounded from above by (d/W(\fracdd+1))^d+1.
  • Sensitivity Analysis for Convex Separable Optimization Over Integral Polymatroids (, and )
    SIAM Journal on Optimization, 28(3):2222–2245, .
    PDF DOI Preliminary version (WINE 2014) Bibtex Abstract
    @article{HarksKlimmPeis2018,
    author = {Tobias Harks and Max Klimm and Britta Peis},
    doi = {10.1137/16M1107450},
    journal = {SIAM Journal on Optimization},
    number = {3},
    pages = {2222--2245},
    title = {Sensitivity Analysis for Convex Separable Optimization Over Integral Polymatroids},
    volume = {28},
    year = {2018},
    }
    We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution, there is a new optimal solution to new parameters such that the $L_1$-norm of the difference of the two solutions is at most two times the $L_1$-norm of the difference of the parameters. We apply these sensitivity results to a class of noncooperative games with a finite set of players where a strategy of a player is to choose a vector in a player-specific integral polymatroid base polytope defined on a common set of elements. The players' private cost functions are regular, convex-separable, and the cost of each element is a nondecreasing function of the own usage of that element and the overall usage of the other players. Under these assumptions, we establish the existence of a pure Nash equilibrium. The existence is proven by an algorithm computing a pure Nash equilibrium that runs in polynomial time whenever the rank of the polymatroid base polytope is polynomially bounded. Both the existence result and the algorithm generalize and unify previous results appearing in the literature. We finally complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any nonpolymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players' strategies are isomorphic to the nonpolymatroid region and that does not admit a pure Nash equilibrium.
  • Bottleneck Routing with Elastic Demands (, and )
    Operations Research Letters, 46(1):93–98, .
    DOI Preliminary version (WINE 2015) Bibtex Abstract
    @article{HarksKlimmSchneider2018,
    author = {Tobias Harks and Max Klimm and Manuel Schneider},
    doi = {10.1016/j.orl.2017.11.007},
    journal = {Operations Research Letters},
    number = {1},
    pages = {93--98},
    title = {Bottleneck Routing with Elastic Demands},
    volume = {46},
    year = {2018},
    }
    Bottleneck routing games are a well-studied model to investigate the impact of selfish behavior in communication networks. In this model, each user selects a path in a network for routing her fixed demand. The disutility of a user only depends on the most congested link visited. We extend this model by allowing users to continuously vary the demand rate at which data is sent along the chosen path. As our main result we establish tight conditions for the existence of pure strategy Nash equilibria.
  • Demand-Independent Optimal Tolls (, and )
    ICALP 2018 – Proc. 45th International Colloquium on Automata, Languages and Programming, pp. 151:1–151:14.
    DOI Bibtex Abstract
    @inproceedings{Colini-BaldeschiKlimmScarsini2018,
    author = {Riccardo Colini{-}Baldeschi and Max Klimm and Marco Scarsini},
    booktitle = {ICALP 2018 – Proc. 45th International Colloquium on Automata, Languages and Programming},
    doi = {10.4230/LIPIcs.ICALP.2018.151},
    editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
    pages = {151:1--151:14},
    series = {LIPIcs},
    title = {Demand-Independent Optimal Tolls},
    volume = {107},
    year = {2018},
    }
    Wardrop equilibria in nonatomic congestion games are in general inefficient as they do not induce an optimal flow that minimizes the total travel time. Network tolls are a prominent and popular way to induce an optimum flow in equilibrium. The classical approach to find such tolls is marginal cost pricing which requires the exact knowledge of the demand on the network. In this paper, we investigate under which conditions demand-independent optimum tolls exist that induce the system optimum flow for any travel demand in the network. We give several characterizations for the existence of such tolls both in terms of the cost structure and the network structure of the game. Specifically we show that demand-independent optimum tolls exist if and only if the edge cost functions are shifted monomials as used by the Bureau of Public Roads. Moreover, non-negative demand-independent optimum tolls exist when the network is a directed acyclic multi-graph. Finally, we show that any network with a single origin-destination pair admits demand-independent optimum tolls that, although not necessarily non-negative, satisfy a budget constraint.
  • Distance-Preserving Graph Contractions (, , , , and )
    ITCS 2018 – Proc. 9th Innovations in Computer Science Conference, pp. 51:1–51:14.
    DOI Extended version (SIDMA 2019) Bibtex Abstract
    @inproceedings{BernsteinDaeubelDisser+2018,
    author = {Aaron Bernstein and Karl D{\"a}ubel and Yann Disser and Max Klimm and Torsten M{\"u}tze and Frieder Smolny},
    booktitle = {ITCS 2018 – Proc. 9th Innovations in Computer Science Conference},
    doi = {10.4230/LIPIcs.ITCS.2018.51},
    editor = {Anna R. Karlin},
    pages = {51:1--51:14},
    series = {LIPIcs},
    title = {Distance-Preserving Graph Contractions},
    volume = {94},
    year = {2018},
    }
    Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding super-vertices remain at distance at least $\phi(d)$ in the contracted graph, where $\varphi$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $\varphi(x) = x/\alpha + \beta$, where $\alpha \geq 1$ and $β \geq 0$ are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of the size of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.
  • Complexity and Approximation of the Continuous Network Design Problem (, and )
    SIAM Journal on Optimization, 27(3):1554–1582, .
    PDF DOI Preliminary version (APPROX 2014) Bibtex
    @article{GairingHarksKlimm2017,
    author = {Martin Gairing and Tobias Harks and Max Klimm},
    doi = {10.1137/15M1016461},
    journal = {SIAM Journal on Optimization},
    number = {3},
    pages = {1554--1582},
    title = {Complexity and Approximation of the Continuous Network Design Problem},
    volume = {27},
    year = {2017},
    abstrat = {We revisit a classical problem in transportation, known as the (bilevel) continuous network design problem, CNDP for short. Given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed, the goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing costs of the induced Wardrop equilibrium and the investment costs for installing the edge's capacities. While this problem is considered to be challenging in the literature, its complexity status was still unknown. We close this gap, showing that CNDP is strongly $\mathsf{NP}$-hard and $\mathsf{APX}$-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions [P. Marcotte, Math. Prog., 34 (1986), pp. 142--162]. We derive a closed form expression of its approximation guarantee for arbitrary sets of latency functions. We then propose a different approximation algorithm and show that it has the same approximation guarantee. Then, we prove that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we derive a closed form expression. For affine latencies, for example, this best-of-two approach achieves an approximation factor of $49/41 \approx 1.195$, which improves on the factor of $5/4$ that has been shown before by Marcotte.},
    }
  • Packing a Knapsack of Unknown Capacity (, , and )
    SIAM Journal on Discrete Mathematics, 31(3):1477–1497, .
    PDF DOI Preliminary version (STACS 2017) Bibtex Abstract
    @article{DisserKlimmMegow+2017,
    author = {Yann Disser and Max Klimm and Nicole Megow and Sebastian Stiller},
    doi = {10.1137/16M1070049},
    journal = {SIAM Journal on Discrete Mathematics},
    number = {3},
    pages = {1477--1497},
    title = {Packing a Knapsack of Unknown Capacity},
    volume = {31},
    year = {2017},
    }
    We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio $\varphi \approx 1.6128$. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items in the beginning and try to pack the items in this order, without changing the order later on. We give efficient algorithms computing these policies. On the other hand, we show that, for any $\alpha > 1$, the problem of deciding whether a given universal policy achieves a factor of $\alpha$ is $\mathsf{coNP}$-complete. If $\alpha$ is part of the input, the same problem is shown to be $\mathsf{coNP}$-complete for items with unit densities. Finally, we show that it is $\mathsf{coNP}$-hard to decide, for given $\alpha$, whether a set of items admits a universal policy with factor $\alpha$, even if all items have unit densities.
  • Impartial Selection and the Power of Up to Two Choices (, and )
    ACM Transactions on Economics and Computation, 5(4):21:1–21:20, .
    PDF DOI Preliminary version (WINE 2015) Bibtex Abstract
    @article{BjeldeFischerKlimm2017,
    author = {Antje Bjelde and Felix A. Fischer and Max Klimm},
    doi = {10.1145/3107922},
    journal = {ACM Transactions on Economics and Computation},
    number = {4},
    pages = {21:1--21:20},
    title = {Impartial Selection and the Power of Up to Two Choices},
    volume = {5},
    year = {2017},
    }
    We study mechanisms that select members of a set of agents based on nominations by other members and that are impartial in the sense that agents cannot influence their own chance of selection. Prior work has shown that deterministic mechanisms for selecting any fixed number $k$ of agents are severely limited and cannot extract a constant fraction of the nominations of the $k$ most highly nominated agents. We prove here that this impossibility result can be circumvented by allowing the mechanism to sometimes but not always select fewer than $k$ agents. This added flexibility also improves the performance of randomized mechanisms, for which we show a separation between mechanisms that make exactly two or up to two choices and give upper and lower bounds for mechanisms allowed more than two choices.
  • MatchTheNet - An Educational Game on 3-Dimensional Polytopes (Multimedia Contribution) (, , and )
    SoCG 2017 – Proc. 33rd International Symposium on Computational Geometry, pp. 66:1–66:5.
    DOI Bibtex Abstract
    @inproceedings{JoswigLohoLorenz+2017,
    author = {Michael Joswig and Georg Loho and Benjamin Lorenz and Rico Raber},
    booktitle = {SoCG 2017 – Proc. 33rd International Symposium on Computational Geometry},
    doi = {10.4230/LIPIcs.SoCG.2017.66},
    editor = {Boris Aronov and Matthew J. Katz},
    pages = {66:1--66:5},
    series = {LIPIcs},
    title = {MatchTheNet - An Educational Game on 3-Dimensional Polytopes (Multimedia Contribution)},
    volume = {77},
    year = {2017},
    }
    We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers.
  • Brief Announcement: Approximation Algorithms for Unsplittable Resource Allocation Problems with Diseconomies of Scale (, and )
    SPAA 2017 – Proc. 29th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 227–229.
    DOI Bibtex Abstract
    @inproceedings{BjeldeKlimmSchmand2017,
    author = {Antje Bjelde and Max Klimm and Daniel Schmand},
    booktitle = {SPAA 2017 – Proc. 29th ACM Symposium on Parallelism in Algorithms and Architectures},
    doi = {10.1145/3087556.3087597},
    editor = {Christian Scheideler and Mohammad Taghi Hajiaghayi},
    pages = {227--229},
    title = {Brief Announcement: Approximation Algorithms for Unsplittable Resource Allocation Problems with Diseconomies of Scale},
    year = {2017},
    }
    We study general resource allocation problems with a diseconomy of scale. Given a finite set of commodities that request certain resources, the cost of each resource grows superlinearly with the demand for it, and our goal is to minimize the total cost of the resources. In large systems with limited coordination, it is natural to consider local dynamics where in each step a single commodity switches its allocated resources whenever the new solution after the switch has smaller total cost over all commodities. This yields a deterministic and polynomial time algorithm with approximation factor arbitrarily close to the locality gap, i.e., the worst case ratio of the cost of a local optimal and a global optimal solution. For costs that are polynomials with non-negative coefficients and maximal degree d, we provide a locality gap for weighted problems that is tight for all values of d. For unweighted problems, the locality gap asymptotically matches the approximation guarantee of the currently best known centralized algorithm [Makarychev, Srividenko FOCS14] but only requires local knowledge of the commodities.
  • Congestion Games with Variable Demands ( and )
    Mathematics of Operations Research, 41(1):255–277, .
    DOI Preliminary version (TARK 2011) Bibtex Abstract
    @article{HarksKlimm2016,
    author = {Tobias Harks and Max Klimm},
    doi = {10.1287/moor.2015.0726},
    journal = {Mathematics of Operations Research},
    number = {1},
    pages = {255--277},
    title = {Congestion Games with Variable Demands},
    volume = {41},
    year = {2016},
    }
    We initiate the study of congestion games with variable demands in which the players strategically choose both a nonnegative demand and a subset of resources. The players' incentives to use higher demands are stimulated by nondecreasing and concave utility functions. The payoff for a player is defined as the difference between the utility of the demand and the associated cost on the used resources. Although this class of noncooperative games captures many elements of real-world applications, it has not been studied in this generality in the past. Specifically, we study the fundamental problem of the existence of pure Nash equilibria, PNE for short. We call a set of cost functions consistent if every congestion game with variable demands and cost functions from the set possesses a PNE. We show that only affine and homogeneous exponential functions are consistent. En route, we obtain novel characterizations of consistency for congestion games with fixed but resource-dependent demands.
  • Undirected Graph Exploration with Θ(log log n) Pebbles (, and )
    SODA 2016 – Proc. 27th ACM-SIAM Symposium on Discrete Algorithms, pp. 25–39.
    DOI Extended version (JACM 2019) Bibtex Abstract
    @inproceedings{DisserHackfeldKlimm2016,
    author = {Yann Disser and Jan Hackfeld and Max Klimm},
    booktitle = {SODA 2016 – Proc. 27th ACM-SIAM Symposium on Discrete Algorithms},
    doi = {10.1137/1.9781611974331.ch3},
    editor = {Robert Krauthgamer},
    pages = {25--39},
    title = {Undirected Graph Exploration with \Θ(log log n) Pebbles},
    year = {2016},
    }
    We consider the fundamental problem of exploring an undirected and initially unknown graph by an agent with little memory. The vertices of the graph are unlabeled, and the edges incident to a vertex have locally distinct labels. In this setting, it is known that $\Theta(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most n vertices. We show that this memory requirement can be decreased significantly by making a part of the memory distributable in the form of pebbles. A pebble is a device that can be dropped to mark a vertex and can be collected when the agent returns to the vertex. We show that for an agent $\mathcal{O}(\log \log n)$ distinguishable pebbles and bits of memory are sufficient to explore any bounded-degree graph with at most n vertices. We match this result with a lower bound exhibiting that for any agent with sub-logarithmic memory, $\Omega(\log \log n)$ distinguishable pebbles are necessary for exploration.
  • Efficiency of Equilibria in Uniform Matroid Congestion Games (, and )
    SAGT 2016 – Proc. 9th International Symposium on Algorithmic Game Theory, pp. 105–116.
    DOI Bibtex Abstract
    @inproceedings{deJongKlimmUetz2016,
    author = {Jasper de Jong and Max Klimm and Marc Uetz},
    booktitle = {SAGT 2016 – Proc. 9th International Symposium on Algorithmic Game Theory},
    doi = {10.1007/978-3-662-53354-3\_9},
    editor = {Martin Gairing and Rahul Savani},
    pages = {105--116},
    series = {LNCS},
    title = {Efficiency of Equilibria in Uniform Matroid Congestion Games},
    volume = {9928},
    year = {2016},
    }
    Network routing games, and more generally congestion games play a central role in algorithmic game theory, comparable to the role of the traveling salesman problem in combinatorial optimization. It is known that the price of anarchy is independent of the network topology for non-atomic congestion games. In other words, it is independent of the structure of the strategy spaces of the players, and for affine cost functions it equals $4/3$. In this paper, we show that the situation is considerably more intricate for atomic congestion games. More specifically, we consider congestion games with affine cost functions where the players' strategy spaces are symmetric and equal to the set of bases of a $k$-uniform matroid. In this setting, we show that the price of anarchy is strictly larger than the price of anarchy for singleton strategy spaces where it is $4/3$. As our main result we show that the price of anarchy can be bounded from above by $28/13 \approx 2.15$. This constitutes a substantial improvement over the price of anarchy bound $5/2$, which is known to be tight for network routing games with affine cost functions.
  • Optimal Impartial Selection ( and )
    SIAM Journal on Computing, 44(5):1263–1285, .
    PDF DOI Preliminary version (EC 2014) Bibtex Abstract
    @article{FischerKlimm2015,
    author = {Felix A. Fischer and Max Klimm},
    doi = {10.1137/140995775},
    journal = {SIAM Journal on Computing},
    number = {5},
    pages = {1263--1285},
    title = {Optimal Impartial Selection},
    volume = {44},
    year = {2015},
    }
    We study a fundamental problem in social choice theory, the selection of a member of a set of agents based on impartial nominations by agents from that set. Studied previously by Alon et al. [Proceedings of TARK, 2011, pp. 101–110] and by Holzman and Moulin [Econometrica, 81 (2013), pp. 173–196], this problem arises when representatives are selected from within a group or when publishing or funding decisions are made based on a process of peer review. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. This is best possible subject to impartiality and resolves a conjecture of Alon et al. Further results are given for the case where some agent receives many nominations and the case where each agent casts at least one nomination.
  • Equilibria in a Class of Aggregative Location Games ( and )
    Journal of Mathematical Economics, 61(1):211–220, .
    DOI Bibtex Abstract
    @article{KlimmHarks2015,
    author = {Max Klimm and Tobias Harks},
    doi = {10.1016/j.jmateco.2015.09.006},
    journal = {Journal of Mathematical Economics},
    number = {1},
    pages = {211--220},
    title = {Equilibria in a Class of Aggregative Location Games},
    volume = {61},
    year = {2015},
    }
    Consider a multimarket oligopoly, where firms have a single license that allows them to supply exactly one market out of a given set of markets. How does the restriction to supply only one market influence the existence of equilibria in the game? To answer this question, we study a general class of aggregative location games where a strategy of a player is to choose simultaneously both a location out of a finite set and a non-negative quantity out of a compact interval. The utility of each player is assumed to depend solely on the chosen location, the chosen quantity, and the aggregated quantity of all other players on the chosen location. We show that each game in this class possesses a pure Nash equilibrium whenever the players' utility functions satisfy the assumptions negative externality, decreasing marginal utility, continuity, and Location–Symmetry. We also provide examples exhibiting that, if one of the assumptions is violated, a pure Nash equilibrium may fail to exist.
  • Computing Network Tolls with Support Constraints (, , and )
    Networks, 65(3):262–285, .
    PDF DOI Bibtex Abstract
    @article{HarksKleinertKlimm+2015,
    author = {Tobias Harks and Ingo Kleinert and Max Klimm and Rolf H. M{\"o}hring},
    doi = {10.1002/net.21604},
    journal = {Networks},
    number = {3},
    pages = {262--285},
    title = {Computing Network Tolls with Support Constraints},
    volume = {65},
    year = {2015},
    }
    Reducing traffic congestion via toll pricing has been a central topic in the operations research and transportation literature and, recently, it has been implemented in several cities all over the world. Since, in practice, it is not feasible to impose tolls on every edge of a given traffic network, we study the resulting mathematical problem of computing tolls on a predefined subset of edges of the network so as to minimize the total travel time of the induced equilibrium flow. We first present an analytical study for the special case of parallel edge networks highlighting the intrinsic complexity and nonconvexity of the resulting optimization problem. We then present algorithms for general networks for which we systematically test the solution quality for large-scale network instances. Finally, we discuss the related optimization problem of computing tolls subject to a cardinality constraint on the number of edges that have tolls.
  • Improving the Hk-Bound on the Price of Stability in Undirected Shapley Network Design Games (, , and )
    Theoretical Computer Science, 562:557–564, .
    PDF DOI Preliminary version (CIAC 2013) Bibtex Abstract
    @article{DisserFeldmannKlimm+2015,
    author = {Yann Disser and Andreas Emil Feldmann and Max Klimm and Mat{\'u}{\v s} Mihal{\'a}k},
    doi = {10.1016/j.tcs.2014.10.037},
    journal = {Theoretical Computer Science},
    pages = {557--564},
    title = {Improving the Hk-Bound on the Price of Stability in Undirected Shapley Network Design Games},
    volume = {562},
    year = {2015},
    }
    In this article we show that the price of stability of Shapley network design games on undirected graphs with $k$ players is at most $\frac{k^3(k+1)/2 - k^2}{1 + k^3(k+1)/2 - k^2}H_k = (1 - \mathcal{O}(1/k^4))H_k$, where $H_k$ denotes the $k$-th harmonic number. This improves on the known upper bound of $H_k$, which is also valid for directed graphs but for these, in contrast, is tight. Hence, we give the first non-trivial upper bound on the price of stability for undirected Shapley network design games that is valid for an arbitrary number of players. Our bound is proved by analyzing the price of stability restricted to Nash equilibria that minimize the potential function of the game. We also present a game with $k=3$ players in which such a restricted price of stability is $1.634$. This shows that the analysis of Bilò and Bove (2011) is tight. In addition, we give an example for three players that improves the lower bound on the (unrestricted) price of stability to $1.571$.
  • Scheduling Bidirectional Traffic on a Path (, and )
    ICALP 2015 – Proc. 42nd International Colloquium on Automata, Languages and Programming, pp. 406–418.
    DOI Bibtex Abstract
    @inproceedings{DisserKlimmLuebecke15,
    author = {Yann Disser and Max Klimm and Elisabeth L{\"u}bbecke},
    booktitle = {ICALP 2015 – Proc. 42nd International Colloquium on Automata, Languages and Programming},
    doi = {10.1007/978-3-662-47672-7\_33},
    editor = {Magn{\'u}s M. Halld{\'o}rsson and Kazuo Iwama and Naoki Kobayashi and Bettina Speckmann},
    pages = {406--418},
    series = {LNCS},
    title = {Scheduling Bidirectional Traffic on a Path},
    volume = {9134},
    year = {2015},
    }
    We study the fundamental problem of scheduling bidirectional traffic along a path composed of multiple segments. The main feature of the problem is that jobs traveling in the same direction can be scheduled in quick succession on a segment, while jobs in opposing directions cannot cross a segment at the same time. We show that this tradeoff makes the problem significantly harder than the related flow shop problem, by proving that it is $\mathsf{NP}$-hard even for identical jobs. We complement this result with a PTAS for a single segment and non-identical jobs. If we allow some pairs of jobs traveling in different directions to cross a segment concurrently, the problem becomes $\mathsf{APX}$-hard even on a single segment and with identical jobs. We give polynomial algorithms for the setting with restricted compatibilities between jobs on a single and any constant number of segments, respectively.
  • Bottleneck Routing with Elastic Demands (, and )
    WINE 2015 – Proc. 11th Conference on Web and Internet Economics, pp. 384–397.
    DOI Extended version (ORL 2018) Bibtex Abstract
    @inproceedings{HarksKlimmSchneider2015,
    author = {Tobias Harks and Max Klimm and Manuel Schneider},
    booktitle = {WINE 2015 – Proc. 11th Conference on Web and Internet Economics},
    doi = {10.1007/978-3-662-48995-6\_28},
    editor = {Evangelos Markakis and Guido Sch{\"a}fer},
    pages = {384--397},
    series = {LNCS},
    title = {Bottleneck Routing with Elastic Demands},
    volume = {9470},
    year = {2015},
    }
    Bottleneck routing games are a well-studied model to investigate the impact of selfish behavior in communication networks. In this model, each user selects a path in a network for routing their fixed demand. The disutility of a used only depends on the most congested link visited. We extend this model by allowing users to continuously vary the demand rate at which data is sent along the chosen path. As our main result we establish tight conditions for the existence of pure strategy Nash equilibria.
  • Impartial Selection and the Power of up to Two Choices (, and )
    WINE 2015 – Proc. 11th Conference on Web and Internet Economics, pp. 146–158.
    DOI Extended version (TEAC 2017) Bibtex Abstract
    @inproceedings{BjeldeFischerKlimm2015,
    author = {Antje Bjelde and Felix A. Fischer and Max Klimm},
    booktitle = {WINE 2015 – Proc. 11th Conference on Web and Internet Economics},
    doi = {10.1007/978-3-662-48995-6\_11},
    editor = {Evangelos Markakis and Guido Sch{\"a}fer},
    pages = {146--158},
    series = {LNCS},
    title = {Impartial Selection and the Power of up to Two Choices},
    volume = {9470},
    year = {2015},
    }
    We study mechanisms that select members of a set of agents based on nominations by other members and that are impartial in the sense that agents cannot influence their own chance of selection. Prior work has shown that deterministic mechanisms for selecting any fixed number of agents are severely limited, whereas randomization allows for the selection of a single agent that in expectation receives at least $1/2$ of the maximum number of nominations. The bound of $1/2$ is in fact best possible subject to impartiality. We prove here that the same bound can also be achieved deterministically by sometimes but not always selecting a second agent. We then show a separation between randomized mechanisms that make exactly two or up to two choices, and give upper and lower bounds on the performance of mechanisms allowed more than two choices.
  • Sharing Non-anonymous Costs of Multiple Resources Optimally ( and )
    CIAC 2015 – Proc. 9th International Conference on Algorithms and Complexity, pp. 274–287.
    DOI Bibtex
    @inproceedings{KlimmSchmand2015,
    author = {Max Klimm and Daniel Schmand},
    booktitle = {CIAC 2015 – Proc. 9th International Conference on Algorithms and Complexity},
    doi = {10.1007/978-3-319-18173-8\_20},
    editor = {Vangelis Th. Paschos and Peter Widmayer},
    pages = {274--287},
    series = {LNCS},
    title = {Sharing Non-anonymous Costs of Multiple Resources Optimally},
    volume = {9079},
    year = {2015abstract=In cost sharing games, the existence and efficiency of pure Nash equilibria fundamentally depends on the method that is used to share the resources' costs. We consider a general class of resource allocation problems in which a set of resources is used by a heterogeneous set of selfish users. The cost of a resource is a (non-decreasing) function of the set of its users. Under the assumption that the costs of the resources are shared by uniform cost sharing protocols, i.e., protocols that use only local information of the resource's cost structure and its users to determine the cost shares, we exactly quantify the inefficiency of the resulting pure Nash equilibria. Specifically, we show tight bounds on prices of stability and anarchy for games with only submodular and only supermodular cost functions, respectively, and an asymptotically tight bound for games with arbitrary set-functions. While all our upper bounds are attained for the well-known Shapley cost sharing protocol, our lower bounds hold for arbitrary uniform cost sharing protocols and are even valid for games with anonymous costs, i.e., games in which the cost of each resource only depends on the cardinality of the set of its users.},
    }
  • Linear, Exponential, but Nothing Else ()
    Chapter in Gems of Combinatorial Optimization and Graph Algorithms (Andreas S. Schulz, Martin Skutella, Sebastian Stiller, Dorothea Wagner, eds.), Springer, .
    DOI Bibtex Abstract
    @incollection{Klimm2015,
    author = {Max Klimm},
    booktitle = {Gems of Combinatorial Optimization and Graph Algorithms},
    doi = {10.1007/978-3-319-24971-1\_11},
    editor = {Andreas S. Schulz and Martin Skutella and Sebastian Stiller and Dorothea Wagner},
    pages = {113--123},
    publisher = {Springer},
    title = {Linear, Exponential, but Nothing Else},
    year = {2015},
    }
    We consider two seemingly unrelated resource allocation problems and show that they share a deep structural property. In the first problem, we schedule jobs on a single machine to minimize the sum of the jobs' cost where each job's cost is determined by a job-specific function of its completion time. In the second problem, we consider weighted congestion games and are interested in the existence of pure Nash equilibria. We show that the classes of delay cost functions for which the scheduling problem admits a priority rule are exactly the classes of resource cost functions that guarantee the existence of a pure Nash equilibrium in weighted congestion games. These classes of cost functions are those that contain only linear functions or exponential functions, but not both.
  • Optimal Impartial Selection ( and )
    EC 2014 – Proc. 15th ACM Conference on Economics and Computation, pp. 803–820.
    PDF DOI Extended version (SICOMP 2015) Bibtex Abstract
    @inproceedings{FischerKlimm2014,
    author = {Felix A. Fischer and Max Klimm},
    booktitle = {EC 2014 – Proc. 15th ACM Conference on Economics and Computation},
    doi = {10.1145/2600057.2602836},
    editor = {Moshe Babaioff and Vincent Conitzer and David A. Easley},
    pages = {803--820},
    title = {Optimal Impartial Selection},
    year = {2014},
    }
    We study the problem of selecting a member of a set of agents based on impartial nominations by agents from that set. The problem was studied previously by Alon et al. and by Holzman and Moulin and has important applications in situations where representatives are selected from within a group or where publishing or funding decisions are made based on a process of peer review. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. Subject to impartiality, this is best possible.
  • Complexity and Approximation of the Continuous Network Design Problem (, and )
    APPROX 2014 – Proc. 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 226–241.
    PDF DOI Extended version (SIOPT 2017) Bibtex Abstract
    @inproceedings{GairingHarksKlimm2014,
    author = {Martin Gairing and Tobias Harks and Max Klimm},
    booktitle = {APPROX 2014 – Proc. 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems},
    doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.226},
    editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
    pages = {226--241},
    series = {LIPIcs},
    title = {Complexity and Approximation of the Continuous Network Design Problem},
    volume = {28},
    year = {2014},
    }
    We revisit a classical problem in transportation, known as the continuous (bilevel) network design problem, CNDP for short. Given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed, the goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing cost of the induced Wardrop equilibrium and the investment cost for installing the capacity. While this problem is considered as challenging in the literature, its complexity status was still unknown. We close this gap showing that CNDP is strongly $\mathsf{NP}$-complete and $\mathsf{APX}$-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions (Math. Program., Vol. 34, 1986). We derive a closed form expression of its approximation guarantee for arbitrary sets of latency functions. We then propose a different approximation algorithm and show that it has the same approximation guarantee. However, we show that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we derive a closed form expression. For affine latencies, e.g., this algorithm achieves a 49/41-approximation which improves on the 5/4 that has been shown before by Marcotte. We finally discuss the case of hard budget constraints on the capacity investment.
  • Approximate Pure Nash Equilibria in Weighted Congestion Games (, and )
    APPROX 2014 – Proc. 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 242–257.
    PDF DOI Bibtex Abstract
    @inproceedings{HansknechtKlimmSkopalik2014,
    author = {Christoph Hansknecht and Max Klimm and Alexander Skopalik},
    booktitle = {APPROX 2014 – Proc. 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems},
    doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.242},
    editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
    pages = {242--257},
    series = {LIPIcs},
    title = {Approximate Pure Nash Equilibria in Weighted Congestion Games},
    volume = {28},
    year = {2014},
    }
    We study the existence of approximate pure Nash equilibria in weighted congestion games and develop techniques to obtain approximate potential functions that prove the existence of $\alpha$-approximate pure Nash equilibria and the convergence of $\alpha$-improvement steps. Specifically, we show how to obtain upper bounds for approximation factor $\alpha$ for a given class of cost functions. For example for concave cost functions the factor is at most $3/2$, for quadratic cost functions it is at most $4/3$, and for polynomial cost functions of maximal degree $\ell$ it is at at most $\ell + 1$. For games with two players we obtain tight bounds which are as small as for example $1.054$ in the case of quadratic cost functions.
  • Packing a Knapsack of Unknown Capacity (, , and )
    STACS 2014 – Proc. 31st International Symposium on Theoretical Aspects of Computer Science, pp. 276–287.
    PDF DOI Extended version (SIDMA 2017) Bibtex Abstract
    @inproceedings{DisserKlimmMegow+2014,
    author = {Yann Disser and Max Klimm and Nicole Megow and Sebastian Stiller},
    booktitle = {STACS 2014 – Proc. 31st International Symposium on Theoretical Aspects of Computer Science},
    doi = {10.4230/LIPIcs.STACS.2014.276},
    editor = {Ernst W. Mayr and Natacha Portier},
    pages = {276--287},
    series = {LIPIcs},
    title = {Packing a Knapsack of Unknown Capacity},
    volume = {25},
    year = {2014},
    }
    We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio $\varphi \approx 1.618$. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items and try to pack the items in this order, independent of the observations made while packing. We give efficient algorithms computing these policies. On the other hand, we show that, for any $\alpha > 1$, the problem of deciding whether a given universal policy achieves a factor of $\alpha$ is $\mathsf{coNP}$-complete. If $\alpha$ is part of the input, the same problem is shown to be $\mathsf{coNP}$-complete for items with unit densities. Finally, we show that it is $\mathsfcoNP-hard to decide, for given $\alpha$, whether a set of items admits a universal policy with factor $\alpha$, even if all items have unit densities.
  • Multimarket Oligopolies with Restricted Market Access ( and )
    SAGT 2014 – Proc. 7th International Symposium on Algorithmic Game Theory, pp. 182–193.
    PDF DOI Bibtex Abstract
    @inproceedings{HarksKlimm2014,
    author = {Tobias Harks and Max Klimm},
    booktitle = {SAGT 2014 – Proc. 7th International Symposium on Algorithmic Game Theory},
    doi = {10.1007/978-3-662-44803-8\_16},
    editor = {Ron Lavi},
    pages = {182--193},
    series = {LNCS},
    title = {Multimarket Oligopolies with Restricted Market Access},
    volume = {8768},
    year = {2014},
    }
    We study the existence of Cournot equilibria in multimarket oligopolies under the additional restriction that every firm may sell its product only to a limited number of markets simultaneously. This situation naturally arises if market entry is subject to a valid license and each firm holds a fixed number of licenses only, or, equivalently, if the firms' short-term assets only suffice to serve up to a certain number of markets. We allow for firm-specific market reaction functions modeling heterogeneity among products. As our main result, we show the existence of a Cournot equilibrium under the following assumptions stated informally below: (i) cost functions are convex; (ii) the marginal return functions strictly decrease for strictly increased own quantities and non-decreased aggregated quantities; (iii) for every firm, the firm-specific price functions across markets are identical up to market-specific shifts. While assumptions (i) and (ii) are frequently imposed in the literature on single market oligopolies, only assumption (iii) seems limiting. We show, however, that if it is violated, there are games without a Cournot equilibrium.
  • Resource Competition on Integral Polymatroids (, and )
    WINE 2014 – Proc. 10th Conference on Web and Internet Economics, pp. 189–202.
    PDF DOI Extended version (SIOPT 2018) Bibtex Abstract
    @inproceedings{HarksKlimmPeis2014,
    author = {Tobias Harks and Max Klimm and Britta Peis},
    booktitle = {WINE 2014 – Proc. 10th Conference on Web and Internet Economics},
    doi = {10.1007/978-3-319-13129-0\_14},
    editor = {Tie{-}Yan Liu and Qi Qi and Yinyu Ye},
    pages = {189--202},
    series = {LNCS},
    title = {Resource Competition on Integral Polymatroids},
    volume = {8877},
    year = {2014},
    }
    We study competitive resource allocation problems in which players distribute their demands integrally over a set of resources subject to player-specific submodular capacity constraints. Each player has to pay for each unit of demand a cost that is a non-decreasing and convex function of the total allocation of that resource. This general model of resource allocation generalizes both singleton congestion games with integer-splittable demands and matroid congestion games with player-specific costs. As our main result, we show that in such general resource allocation problems a pure Nash equilibrium is guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure Nash equilibrium.
  • Congestion Games with Higher Demand Dimensions ( and )
    WINE 2014 – Proc. 10th Conference on Web and Internet Economics, pp. 453–459.
    PDF DOI Extended version (MOR 2022) Bibtex Abstract
    @inproceedings{KlimmSchuetz2014,
    author = {Max Klimm and Andreas Sch{\"u}tz},
    booktitle = {WINE 2014 – Proc. 10th Conference on Web and Internet Economics},
    doi = {10.1007/978-3-319-13129-0\_39},
    editor = {Tie{-}Yan Liu and Qi Qi and Yinyu Ye},
    pages = {453--459},
    series = {LNCS},
    title = {Congestion Games with Higher Demand Dimensions},
    volume = {8877},
    year = {2014},
    }
    We introduce a generalization of weighted congestion games in which players are associated with $k$-dimensional demand vectors and resource costs are k-dimensional functions $c : \mathbb{R}^k_{\geq 0} \to \mathbb{R}$ of the aggregated demand vector of the players using the resource. Such a cost structure is natural when the cost of a resource depends on different properties of the players' demands, e.g., total weight, total volume, and total number of items. A complete characterization of the existence of pure Nash equilibria in terms of the resource cost functions for all $k \in \mathbb{N}$ is given.
  • Computing Pure Nash and Strong Equilibria in Bottleneck Congestion Games (, , and )
    Mathematical Programming, 141(1-2):193–215, .
    PDF DOI Preliminary version (ESA 2010) Bibtex Abstract
    @article{HarksHoeferKlimm+2013,
    author = {Tobias Harks and Martin Hoefer and Max Klimm and Alexander Skopalik},
    doi = {10.1007/s10107-012-0521-3},
    journal = {Mathematical Programming},
    number = {1-2},
    pages = {193--215},
    title = {Computing Pure Nash and Strong Equilibria in Bottleneck Congestion Games},
    volume = {141},
    year = {2013},
    }
    Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria—a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly $\mathsf{NP}$-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria.
  • Strong Equilibria in Games with the Lexicographical Improvement Property (, and )
    International Journal of Game Theory, 42(2):461–482, .
    PDF DOI Preliminary version (WINE 2009) Bibtex Abstract
    @article{HarksKlimmMoehring2013,
    author = {Tobias Harks and Max Klimm and Rolf H. M{\"o}hring},
    doi = {10.1007/s00182-012-0322-1},
    journal = {International Journal of Game Theory},
    number = {2},
    pages = {461--482},
    title = {Strong Equilibria in Games with the Lexicographical Improvement Property},
    volume = {42},
    year = {2013},
    }
    We study a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the lexicographical improvement property (LIP) and show that, in finite games, it is equivalent to the existence of a generalized strong potential function. We use this characterization to derive existence, efficiency and fairness properties of strong equilibria (SE). As our main result, we show that an important class of games that we call bottleneck congestion games has the LIP and thus the above mentioned properties. For infinite games, the LIP does neither imply the existence of a generalized strong potential nor the existence of SE. We therefore introduce the slightly more general concept of the pairwise LIP and prove that whenever the pairwise LIP is satisfied for a continuous function, then there exists a SE. As a consequence, we show that splittable bottleneck congestion games with continuous facility cost functions possess a SE.
  • Improving the Hk-Bound on the Price of Stability in Undirected Shapley Network Design Games (, , and )
    CIAC 2013 – Proc. 8th International Conference on Algorithms and Complexity, pp. 158–169.
    PDF DOI Extended version (TCS 2015) Bibtex Abstract
    @inproceedings{DisserFeldmannKlimm+2013,
    author = {Yann Disser and Andreas Emil Feldmann and Max Klimm and Mat{\'u}{\v s} Mihal{\'a}k},
    booktitle = {CIAC 2013 – Proc. 8th International Conference on Algorithms and Complexity},
    doi = {10.1007/978-3-642-38233-8\_14},
    editor = {Paul G. Spirakis and Maria J. Serna},
    pages = {158--169},
    series = {LNCS},
    title = {Improving the Hk-Bound on the Price of Stability in Undirected Shapley Network Design Games},
    volume = {7878},
    year = {2013},
    }
    In this paper we show that the price of stability of Shapley network design games on undirected graphs with $k$ players is at most $\frac{k^3(k+1)/2 - k^2}{1 + k^3(k+1)/2 - k^2}H_k = (1 - \mathcal{O}(1/k^4))H_k$, where $H_k$ denotes the $k$-th harmonic number. This improves on the known upper bound of $H_k$, which is also valid for directed graphs but for these, in contrast, is tight. Hence, we give the first non-trivial upper bound on the price of stability for undirected Shapley network design games that is valid for an arbitrary number of players. Our bound is proved by analyzing the price of stability restricted to Nash equilibria that minimize the potential function of the game. We also present a game with $k=3$ players in which such a restricted price of stability is $1.634$. This shows that the analysis of Bilò and Bove (Journal of Interconnection Networks, Volume 12, 2011) is tight. In addition, we give an example for three players that improves the lower bound on the (unrestricted) price of stability to $1.571$.
  • Congestion Games with Player-Specific Costs Revisited ( and )
    SAGT 2013 – Proc. 6th International Symposium on Algorithmic Game Theory, pp. 98–109.
    PDF DOI Bibtex Abstract
    @inproceedings{GairingKlimm2013,
    author = {Martin Gairing and Max Klimm},
    booktitle = {SAGT 2013 – Proc. 6th International Symposium on Algorithmic Game Theory},
    doi = {10.1007/978-3-642-41392-6\_9},
    editor = {Berthold V{\"o}cking},
    pages = {98--109},
    series = {LNCS},
    title = {Congestion Games with Player-Specific Costs Revisited},
    volume = {8146},
    year = {2013},
    }
    We study the existence of pure Nash equilibria in congestion games with player-specific costs. Specifically, we provide a thorough characterization of the maximal sets of cost functions that guarantee the existence of a pure Nash equilibrium. For the case that the players are unweighted, we show that it is necessary and sufficient that for every resource and for every pair of players the corresponding cost functions are affine transformations of each other. For weighted players, we show that in addition one needs to require that all cost functions are affine or all cost functions are exponential. Finally, we construct a four-player singleton weighted congestion game where the cost functions are identical among the resources and differ only by an additive constant among the players and show that it does not have a pure Nash equilibrium. This answers an open question by Mavronicolas et al. who showed that such games with at most three players always have a pure Nash equilibrium.
  • On the Existence of Pure Nash Equilibria in Weighted Congestion Games ( and )
    Mathematics of Operations Research, 37(3):419–436, .
    PDF DOI Preliminary version (ICALP 2010) Bibtex Abstract
    @article{HarksKlimm2012,
    author = {Tobias Harks and Max Klimm},
    doi = {10.1287/moor.1120.0543},
    journal = {Mathematics of Operations Research},
    number = {3},
    pages = {419--436},
    title = {On the Existence of Pure Nash Equilibria in Weighted Congestion Games},
    volume = {37},
    year = {2012},
    }
    We study the existence of pure Nash equilibria in weighted congestion games. Let $\mathcal{C}$ denote a set of cost functions. We say that $\mathcal{C}$ is consistent if every weighted congestion game with cost functions in $\mathcal{C}$ possesses a pure Nash equilibrium. Our main contribution is a complete characterization of consistency of continuous cost functions. We prove that a set $\mathcal{C}$ of continuous functions is consistent for two-player games if and only if $\mathcal{C}$ contains only monotonic functions and for all nonconstant functions $c_1, c_2 \in \mathcal{C}$, there are constants $a, b \in \mathbb{R}$ such that $c_1(x) = a c_2(x) + b$ for all $x \in \mathbb{R}_{\geq 0}$. For games with at least three players, we prove that $\mathcal{C}$ is consistent if and only if exactly one of the following cases holds: (a) $\mathcal{C}$ contains only affine functions; (b) $\mathcal{C}$ contains only exponential functions such that $c(x) = a_c e^{\phi x} + b_c$ for some $a_c, b_c, \phi ∈ \mathbb{R}$, where $a_c$ and $b_c$ may depend on $c$, while $\phi$ must be equal for every $c \in \mathcal{C}$. The latter characterization is even valid for three-player games. Finally, we derive several characterizations of consistency of cost functions for games with restricted strategy spaces, such as weighted network congestion games or weighted congestion games with singleton strategies.
  • Conflict-Free Vehicle Routing (, , and )
    EURO Journal on Transportation and Logistics, 1(1-2):87–111, .
    DOI Bibtex Abstract
    @article{GawrilowKlimmMoehring+2012,
    author = {Ewgenij Gawrilow and Max Klimm and Rolf H. M{\"o}hring and Bj{\"o}rn Stenzel},
    doi = {10.1007/s13676-012-0008-7},
    journal = {EURO Journal on Transportation and Logistics},
    number = {1-2},
    pages = {87--111},
    title = {Conflict-Free Vehicle Routing},
    volume = {1},
    year = {2012},
    }
    Collision-free vehicle routing occurs in many applications from robot movements via scheduling multiple cranes in steel logistics to the transport of containers by automated guided vehicles. In cooperation with the HHLA Container Terminal Altenwerder (CTA), we have developed a dynamic online routing algorithm that computes collision-free routes for AGVs by considering implicit time-expanded networks. This approach proved to be very efficient in scenarios with a high traffic density. This is in contrast to the more frequent static approaches that use routes computed in the static graph—i.e., without time expansion—and employ additional methods for collision avoidance during execution of the static routes. These static approaches have the advantage that they are quite robust against disturbances but are rather unpredictable because of the usually heuristic collision avoidance rules that may even run into deadlocks in high traffic scenarios. In this paper, we study if the static approach can be suitably enhanced to meet the performance of the dynamic approach and become predictable and collision and deadlock-free. Our approach is based on online static route computations combined with load balancing techniques and graph algorithms for guaranteed deadlock avoidance. We evaluate our static algorithm on routing scenarios from the HHLA CTA. It turns out that the performance of our static router is slightly superior in low and medium traffic scenarios, but loses against the dynamic router in high traffic scenarios.
  • Characterizing the Existence of Potential Functions in Weighted Congestion Games (, and )
    Theory of Computing Systems, 49(1):46–70, .
    PDF DOI Preliminary version (SAGT 2009) Bibtex Abstract
    @article{HarksKlimmMoehring2011,
    author = {Tobias Harks and Max Klimm and Rolf H. M{\"o}hring},
    doi = {10.1007/s00224-011-9315-x},
    journal = {Theory of Computing Systems},
    number = {1},
    pages = {46--70},
    title = {Characterizing the Existence of Potential Functions in Weighted Congestion Games},
    volume = {49},
    year = {2011},
    }
    Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let $\mathcal{C}$ be an arbitrary set of locally bounded functions and let $\mathcal{G}(\mathcal{C})$ be the set of weighted congestion games with cost functions in $\mathcal{C}$. We show that every weighted congestion game $G \in \mathcal{G}(\mathcal{C})$ admits an exact potential if and only if $\mathcal{C}$ contains only affine functions. We also give a similar characterization for $w$-potentials with the difference that here $\mathcal{C}$ consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facility-dependent demands and elastic demands, respectively.
  • Congestion Games with Variable Demands ( and )
    TARK 2011 – Proc. 13th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 111–120.
    DOI Extended version (MOR 2016) Bibtex Abstract
    @inproceedings{HarksKlimm2011a,
    author = {Tobias Harks and Max Klimm},
    booktitle = {TARK 2011 – Proc. 13th Conference on Theoretical Aspects of Rationality and Knowledge},
    doi = {10.1145/2000378.2000391},
    editor = {Krzysztof R. Apt},
    pages = {111--120},
    title = {Congestion Games with Variable Demands},
    year = {2011},
    }
    We initiate the study of congestion games with variable demands where the (variable) demand has to be assigned to exactly one subset of resources. The players' incentives to use higher demands are stimulated by non-decreasing and concave utility functions. The payoff for a player is defined as the difference between the utility of the demand and the associated cost on the used resources. Although this class of non-cooperative games captures many elements of real-world applications, it has not been studied in this generality, to our knowledge, in the past. We study the fundamental problem of the existence of pure Nash equilibria (PNE for short) in congestion games with variable demands. We call a set of cost functions $C$ consistent if every congestion game with variable demands and cost functions in $C$ possesses a PNE. We say that $C$ is FIP consistent if every such game possesses the $\alpha$-Finite Improvement Property for every $\alpha > 0$. Our main results provide a complete characterization of consistency of cost functions revealing structural differences to congestion games with fixed demands (weighted congestion games), where in the latter even inhomogeneously exponential functions are FIP consistent. Finally, we study consistency and FIP consistency of cost functions in a slightly different class of games, where every player experiences the same cost on a resource (uniform cost model). We give a characterization of consistency and FIP consistency showing that only homogeneously exponential functions are consistent while no functions are FIP consistent.
  • Demand Allocation Games: Integrating Discrete and Continuous Strategy Spaces ( and )
    WINE 2011 – Proc. 7th Conference on Web and Internet Economics, pp. 194–205.
    DOI Bibtex Abstract
    @inproceedings{HarksKlimm2011b,
    author = {Tobias Harks and Max Klimm},
    booktitle = {WINE 2011 – Proc. 7th Conference on Web and Internet Economics},
    doi = {10.1007/978-3-642-25510-6\_17},
    editor = {Ning Chen and Edith Elkind and Elias Koutsoupias},
    pages = {194--205},
    series = {LNCS},
    title = {Demand Allocation Games: Integrating Discrete and Continuous Strategy Spaces},
    volume = {7090},
    year = {2011},
    }
    In this paper, we introduce a class of games which we term demand allocation games that combines the characteristics of finite games such as congestion games and continuous games such as Cournot oligopolies. In a strategy profile each player may choose both an action out of a finite set and a non-negative demand out of a convex and compact interval. The utility of each player is assumed to depend solely on the action, the chosen demand, and the aggregated demand on the action chosen. We show that this general class of games possess a pure Nash equilibrium whenever the players' utility functions satisfy the assumptions negative externality, decreasing marginal returns and homogeneity. If one of the assumptions is violated, then a pure Nash equilibrium may fail to exist. We demonstrate the applicability of our results by giving several concrete examples of games that fit into our model.
  • Optimal File Distribution in Peer-to-Peer Networks (, , and )
    ISAAC 2011 – Proc. 22nd International Symposium on Algorithms and Computation, pp. 210–219.
    DOI Extended version (J. Sched. 2020) Bibtex Abstract
    @inproceedings{GoetzmannHarksKlimm+2011,
    author = {Kai{-}Simon Goetzmann and Tobias Harks and Max Klimm and Konstantin Miller},
    booktitle = {ISAAC 2011 – Proc. 22nd International Symposium on Algorithms and Computation},
    doi = {10.1007/978-3-642-25591-5\_23},
    editor = {Takao Asano and Shin{-}Ichi Nakano and Yoshio Okamoto and Osamu Watanabe},
    pages = {210--219},
    series = {LNCS},
    title = {Optimal File Distribution in Peer-to-Peer Networks},
    volume = {7074},
    year = {2011},
    }
    We study the problem of distributing a file initially located at a server among a set of peers. Peers who downloaded the file can upload it to other peers. The server and the peers are connected to each other via a core network. The upload and download rates to and from the core are constrained by user and server specific upload and download capacities. Our objective is to minimize the makespan. We derive exact polynomial time algorithms for the case when upload and download capacities per peer and among peers are equal. We show that the problem becomes strongly NP-hard for equal upload and download capacities per peer that may differ among peers. For this case we devise a polynomial time $(1 + 2\sqrt{2})$-approximation algorithm. To the best of our knowledge, neither $\mathsf{NP}$-hardness nor approximation algorithms were known before for this problem.
  • On the Existence of Pure Nash Equilibria in Weighted Congestion Games ( and )
    ICALP 2010 – Proc. 37th International Colloquium on Automata, Languages and Programming, pp. 79–89.
    DOI Extended version (MOR 2012) Bibtex Abstract
    @inproceedings{HarksKlimm2010,
    author = {Tobias Harks and Max Klimm},
    booktitle = {ICALP 2010 – Proc. 37th International Colloquium on Automata, Languages and Programming},
    doi = {10.1007/978-3-642-14165-2\_8},
    editor = {S. Abramsky and C. Gavoille and C. Kirchner and F. {Meyer auf der Heide} and P. G. Spirakis},
    pages = {79--89},
    series = {LNCS},
    title = {On the Existence of Pure Nash Equilibria in Weighted Congestion Games},
    volume = {6198},
    year = {2010},
    }
    We study the existence of pure Nash equilibria in weighted congestion games. Let $\mathcal{C}$ denote a set of cost functions. We say that $\mathcal{C}$ is consistent if every weighted congestion game with cost functions in $\mathcal{C}$ possesses a pure Nash equilibrium. We say that $\mathcal{C}$ is FIP-consistent if every weighted congestion game with cost functions in $\mathcal{C}$ has the Finite Improvement Property. Our main results are structural characterizations of consistency for twice continuously differentiable cost functions. More specifically, we show that $\mathcal{C}$ is consistent for two-player games if and only if $\mathcal{C}$ contains only monotonic functions and for all $c_1, c_2 \in \mathcal{C}$, there are constants $a, b \in \mathbb{R}$ such that $c_1 = a c_2 + b$. For games with at least 3 players we show that $\mathcal{C}$ is consistent if and only if exactly one of the following cases hold: (i) $\mathcal{C}$ contains only affine functions; (ii) $\mathcal{C}$ contains only exponential functions such that $c(\ell) = a_c e^{\phi \ell} + b_c$ for some $a_c, b_c, \phi \in \mathbb{R}$, where $a_c$ and $b_c$ may depend on $c$, while $\phi$ must be equal for every $c \in \mathcal{C}$. This characterization is even valid for 3-player games, thus, closing the gap to 2-player games considered above. Finally, we derive various results regarding consistency and FIP-consistency for weighted network congestion games.
  • Computing Pure Nash and Strong Equilibria in Bottleneck Congestion Games (, , and )
    ESA 2010 – Proc. 18th European Symposium on Algorithms, pp. 29–38.
    DOI Extended version (Math. Program. (2013) Bibtex Abstract
    @inproceedings{HarksHoeferKlimm+2010,
    author = {Tobias Harks and Martin Hoefer and Max Klimm and Alexander Skopalik},
    booktitle = {ESA 2010 – Proc. 18th European Symposium on Algorithms},
    doi = {10.1007/978-3-642-15781-3\_3},
    editor = {Mark de Berg and Ulrich Meyer},
    pages = {29--38},
    series = {LNCS},
    title = {Computing Pure Nash and Strong Equilibria in Bottleneck Congestion Games},
    volume = {6347},
    year = {2010},
    }
    Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria – a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly $\mathsf{NP}$-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria.
  • Characterizing the Existence of Potential Functions in Weighted Congestion Games (, and )
    SAGT 2009 – Proc. 2nd International Symposium on Algorithmic Game Theory, pp. 97–108.
    DOI Extended version (TOCS 2011) Bibtex Abstract
    @inproceedings{HarksKlimmMoehring2009b,
    author = {Tobias Harks and Max Klimm and Rolf H. M{\"o}hring},
    booktitle = {SAGT 2009 – Proc. 2nd International Symposium on Algorithmic Game Theory},
    doi = {10.1007/978-3-642-04645-2\_10},
    editor = {Marios Mavronicolas and Vicky G. Papadopoulou},
    pages = {97--108},
    series = {LNCS},
    title = {Characterizing the Existence of Potential Functions in Weighted Congestion Games},
    volume = {5814},
    year = {2009},
    }
    Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let $\mathcal{C}$ be an arbitrary set of locally bounded functions and let $\mathcal{G}(\mathcal{C})$ be the set of weighted congestion games with cost functions in $\mathcal{C}$. We show that every weighted congestion game $G \in \mathcal{G}(\mathcal{C})$ admits an exact potential if and only if $\mathcal{C}$ contains only affine functions. We also give a similar characterization for weighted potentials with the difference that here $\mathcal{C}$ consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facility-dependent demands and elastic demands, respectively.
  • Strong Nash Equilibria in Games with the Lexicographical Improvement Property (, and )
    WINE 2009 – Proc. 5th Conference on Web and Internet Economics, pp. 463–470.
    DOI Extended version (IJGT 2013) Bibtex Abstract
    @inproceedings{HarksKlimmMoehring2009a,
    author = {Tobias Harks and Max Klimm and Rolf H. M{\"o}hring},
    booktitle = {WINE 2009 – Proc. 5th Conference on Web and Internet Economics},
    doi = {10.1007/978-3-642-10841-9\_43},
    editor = {Stefano Leonardi},
    pages = {463--470},
    series = {LNCS},
    title = {Strong {Nash} Equilibria in Games with the Lexicographical Improvement Property},
    volume = {5929},
    year = {2009},
    }
    We provide an axiomatic framework for the the well studied lexicographical improvement property and derive new results on the existence of strong Nash equilibria for a very general class of congestion games with bottleneck objectives. This includes extensions of classical load-based models, routing games with splittable demands, scheduling games with malleable jobs, and more.