Fast Algorithms for the Undirected Negative Cost Cycle Detection Problem

The undirected negative cost cycle detection (UNCCD) problem is concerned with checking whether an undirected, weighted graph contains a negative cost cycle. Known approaches for solving this problem involve reducing it to either the minimum weight$$b$$b-matching problem or the minimum weight$$T$$T-join problem. In this paper, we formally describe these two approaches and provide the tightest known analysis of the $$b$$b-matching approach. Our analysis establishes that the $$b$$b-matching approach runs in $$O((n+m)^{2}\cdot \log (n+m))$$O((n+m)2·log(n+m)) time on a graph with $$n$$n nodes and $$m$$m edges. We also explore the case where the edge costs are integers in the range $$\{-K,\ldots , K\}$${-K,…,K}. For $$K = O(1)$$K=O(1), we provide improved time bounds for both the above-mentioned approaches by exploiting the existence of specialized procedures for optimization problems in graphs with integral positive edge costs. We show that while the $$T$$T-join algorithm has a better time bound for general graphs, the $$b$$b-matching algorithm is more efficient for sparse graphs. Finally, we present one of the first, extensive empirical studies for the UNCCD problem. We examine both approaches on several different families of undirected graphs. Our findings provide insight into the actual efficiency of each approach and reinforce the asymptotic analysis of our algorithms.