Dr. Philipp Warode

 

Now at: Humboldt-Universität zu Berlin

Email: philipp.warode@hu-berlin.de

0000-0002-2878-6872
Bibliography on dblp
Google Scholar
GitHub

Publications

to appear

• Optimizing Throughput and Makespan of Queuing Systems by Information Design (Svenja M. Griesbach, Max Klimm, Philipp Warode, and Theresa Ziemke)
  ESA 2024 – Proc. 32nd European Symposium on Algorithms.
  Bibtex Abstract
  @inproceedings{GriesbachKlimmWarode+2024,
  author = {Svenja M. Griesbach and Max Klimm and Philipp Warode and Theresa Ziemke},
  title = {Optimizing Throughput and Makespan of Queuing Systems by Information Design},
  booktitle = {ESA 2024 – Proc. 32nd European Symposium on Algorithms},
  year = {to appear},
  }
  
  We study the optimal provision of information for two natural performance measures of queuing systems: throughput and makespan. A set of parallel links (queues) is equipped with deterministic capacities and stochastic travel times where the latter depend on a realized scenario, and the number of scenarios is assumed to be constant. A continuum of flow particles (users) arrives at the system at a constant rate. A system operator knows the realization of the scenario and may (partially) reveal this information via a public signaling scheme to the flow particles. Upon arrival, the flow particles observe the signal issued by the system operator, form an updated belief about the realized scenario, and decide on a link to use. Inflow into a link exceeding the link's capacity builds up in a queue that increases the travel time on the link. Dynamic inflow rates are in a Bayesian dynamic equilibrium when the expected travel time along all links with positive inflow is equal at every point in time. For a given time horizon $T$, the throughput induced by a signaling scheme is the total volume of flow that leaves the links in the interval $[0,T]$. We show that the public signaling scheme that maximizes the throughput may involve irrational numbers and provide an additive polynomial time approximation scheme (PTAS) that approximates the optimal throughput by an arbitrary additive constant $\epsilon>0$. The algorithm solves a Langrangian dual of the signaling problem with the Ellipsoid method whose separation oracle is implemented by a cell decomposition technique. We also provide a multiplicative fully polynomial time approximation scheme (FPTAS) that does not rely on strong duality and, thus, allows to compute also the optimal signals. It uses a different cell decomposition technique together with a piece-wise convex under-estimator of the optimal value function. Finally, we consider the makespan of a Bayesian dynamic equilibrium which is defined as the last point in time when a total given value of flow leaves the system. Using a variational inequality argument, we show that full information revelation is a public signaling scheme that minimizes the makespan.

2022

• Parametric Computation of Minimum Cost Flows with Piecewise Quadratic Costs (Max Klimm and Philipp Warode)
  Mathematics of Operations Research, 47(1):812–846, 2022.
  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.
• Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling (Sven Jäger, Guillaume Sagnol, Daniel Schmidt genannt Waldschmidt, and Philipp Warode)
  Preprint, 2022.
  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.

2021

• Approximate Parametric Computation of Minimum-Cost Flows with Convex Costs (Per Joachims, Max Klimm, and Philipp Warode)
  Preprint, 2021.
  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.

2020

• Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians (Max Klimm and Philipp Warode)
  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.

2019

• Computing all Wardrop Equilibria parametrized by the Flow Demand (Max Klimm and Philipp Warode)
  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.

Thesis

• Parametric Computation of Equilibria and Flows (Philipp Warode)
  PhD thesis, 2022.
  pdf doi

Software

• paminco (Per Joachims, Max Klimm and Philipp Warode)
  A Python package for the parametric computation of minimum cost flows.
  docs source