IWOCA 2015: Combinatorial Algorithms pp 25-37

A Fast Scaling Algorithm for the Weighted Triangle-Free 2-Matching Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9538)

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 $$O(\sqrt{V}E \log {V} \log (VC))$$.

Keywords

Parallel Edge Edge Cost Free Vertex Maximal Prefix Suitable Implementation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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