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

## TL;DR

This paper introduces new algorithms for finding the minimum cost-to-time ratio cycle in directed graphs, improving efficiency especially for sparse graphs and graphs with certain structures, with applications in optimization and verification.

## Contribution

It presents the first improvement over Megiddo's classic algorithm for sparse graphs and introduces parallel algorithms for negative cycle detection and shortest paths.

## Key findings

- Achieved a running time of  O(m^{3/4} n^{3/2}) for sparse graphs
- Developed an algorithm with n^3 / 2^{\u03a9(\u221a{\log n})} time for general graphs
- Created an  O(n) time algorithm for graphs with constant treewidth

## 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.