Parametric Computation of Minimum Cost Flows with Piecewise Quadratic Costs (Max Klimm and Philipp Warode)
Mathematics of Operations Research, 47(1):812–846, 2022.
@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.
@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.