Negative Cycles in Weighted Digraphs

Years and Authors of Summarized Original Work

• 1994; Kavvadias, Pantziou, Spirakis, Zaroliagis

Problem Definition

Let ${ G=(V,E) }$ be an n-vertex, m-edge directed graph (digraph), whose edges are associated with a real-valued cost function ${ \textit{w}t:E\to \mathbb{R}}$. The cost, ${ \textit{w}t(P) }$, of a path P in G is the sum of the costs of the edges of P. A simple path C whose starting and ending vertices coincide is called a cycle. If ${ \textit{w}t(C)<0 }$, then C is called a negative cycle. The goal of the negative cycle problem is to detect whether there is such a cycle in a given digraph G with real-valued edge costs, and if indeed exists to output the cycle.   

The negative cycle problem is closely related to the shortest path problem. In the latter, a minimum cost path between two vertices s and t is sought. It is easy to see that an s-t shortest path exists if and only if no s-t path in G contains a negative cycle [1, 13]. It is also well-known that...