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arXiv · 1704.08122

Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs

Karl Bringmann·Thomas Dueholm Hansen·Sebastian Krinninger
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Abstract

We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with $ n $ nodes and $ m $ edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time $ \tilde O (m^{3/4} n^{3/2}) $, which gives the first improvement over Megiddo's $ \tilde O (n^3) $ algorithm [JACM'83] for sparse graphs. We further demonstrate how to obtain both an algorithm with running time $ n^3 / 2^{\Omega{(\sqrt{\log n})}} $ on general graphs and an algorithm with running time $ \tilde O (n) $ on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest.

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Karl Bringmann, Thomas Dueholm Hansen, Sebastian Krinninger. 2017-04-26. Improved Algorithms for Computing the Cycle of Minimum Cost-to-Time Ratio in Directed Graphs. https://doi.org/10.4230/lipics.icalp.2017.124

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