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

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.