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 $x :E \rightarrow \left\{ 0,1,2 \right\}$ such that for each node $v \in V$ the sum of values x(e) on all edges e incident to v equals 2. If $\mathop {\text {supp}}\nolimits (x) = \left\{ e \in E \mid x(e) \ne 0 \right\}$ contains no triangles, then x is called triangle-free. Polyhedrally, triangle-free 2-matchings are harder than 2-matchings, but easier than usual 1-matchings.

Concerning the weighted case, Cornuéjols and Pulleyblank devised a combinatorial strongly-polynomial algorithm that finds a perfect triangle-free 2-matching of minimum cost. A suitable implementation of their algorithm runs in $O(VE \log V)$ time.

In case of integer edge costs in the range [0, C], for both 1- and 2-matchings some faster scaling algorithms are known that find optimal solutions within $O(\sqrt{V\alpha (E, V)\log {V}}E \log (VC))$ and $O(\sqrt{V}E \log (VC))$ time, respectively, where $\alpha$ denotes the inverse Ackermann function. So far, no efficient cost-scaling algorithm is known for finding a minimum-cost perfect triangle-free 2-matching. The present paper fills this gap by presenting such an algorithm with time complexity of $O(\sqrt{V}E \log {V} \log (VC))$.