Improved Algorithms for Detecting Negative Cost Cycles in Undirected Graphs

In this paper, we explore the design of algorithms for the problem of checking whether an undirected graph contains a negative cost cycle (UNCCD). It is known that this problem is significantly harder than the corresponding problem in directed graphs. Current approaches for solving this problem involve reducing it to either the b-matching problem or the T -join problem. The latter approach is more efficient in that it runs in O(n) time on a graph with n vertices and m edges, while the former runs in O(n) time. This paper shows that instances of the UNCCD problem, in which edge weights are restricted to be in the range {−K · ·K} can be solved in O(n · log n) time. Our algorithm is basically a variation of the T -join approach, which exploits the existence of extremely efficient shortest path algorithms in graphs with integral positive weights. We also provide an implementation profile of the algorithms discussed.