A fast scaling algorithm for the weighted triangle-free 2-matching problem

Abstract

A perfect 2-matching in an undirected graph G=(V,E) is a function x:E→0,1,2 such that for each node v∈V the sum of values x(e) on all edges e incident to v equals 2. If supp(x)=e∈E∣x(e)≠0 contains no triangles then x is called triangle-free.Polyhedrally speaking, triangle-free 2-matchings are harder than 2-matchings, but easier than usual 1-matchings.Given edge costs c:E→R+, a natural combinatorial problem consists in finding a perfect triangle-free matching of minimum total cost. For this problem, Cornuéjols and Pulleyblank devised a combinatorial strongly-polynomial algorithm, which can be implemented to run in O(VElogV) time. (Here we write V, E to indicate their cardinalities |V|, |E|.)If edge costs are integers in range [0,C] then for both 1- and 2-matchings some faster scaling algorithms are known that find optimal solutions within O(Vα(E,V)logVElog(VC)) and O(VElog(VC)) time, respectively, where α denotes the inverse Ackermann function. So far, no efficient cost-scaling algorithm is known for finding a minimum-cost perfect triangle-free2-matching. The present paper fills this gap by presenting such an algorithm with time complexity of O(VElogVlog(VC)).