Exact Solution Algorithms for the Chordless Cycle Problem

Abstract

A formulation, a heuristic, and branch-and-cut algorithms are investigated for the chordless cycle problem. This is the problem of finding a largest simple cycle for a given graph so that no edge between nonimmediately subsequent cycle vertices is contained in the graph. Leaving aside procedures based on complete enumeration, no previous exact solution algorithm appears to exist for the problem, which is relevant both in theoretical and practical terms. Extensive computational results are reported here for randomly generated graphs and for graphs originating from the literature. Under acceptable CPU times, certified optimal solutions are presented for graphs with as many as 100 vertices. Summary of Contribution: Finding chordless cycles of a graph, also known as holes, is relevant, among others, to graph theory, to the design of polyhedral based exact solution algorithms to integer programming (IP) problems, and to the practical applications that benefit from these algorithms. For instance, perfect graphs do not contain odd holes. Additionally, odd hole inequalities are valid for strengthening the formulations to numerous problems that are directly defined over graphs. Furthermore, these inequalites, in association with applicable conflict graphs, are used by all modern IP solvers to preprocess and strengthen virtually any IP formulation submitted to them.