Faster Algorithm for Finding Maximum 1-Restricted Simple 2-Matchings

Abstract

We revisit the problem of finding a 1-restricted simple 2-matching of maximum cardinality. Recall that, given an undirected graph ${\varvec{G = (V, E)}}$, a simple 2-matching is a subset ${\varvec{M}} \subseteq {\varvec{E}}$ of edges such that each node in ${\varvec{V}}$ is incident to at most two edges in ${\varvec{M}}$. Clearly, each such ${\varvec{M}}$ decomposes into a node-disjoint collection of paths and circuits. ${\varvec{M}}$ is called 1-restricted if it contains no isolated edges (i.e. paths of length one). A combinatorial polynomial algorithm for finding such ${\varvec{M}}$ of maximum cardinality and also a min-max relation were devised by Hartvigsen. It was shown that finding such ${\varvec{M}}$ amounts to computing a (not necessarily 1-restricted) simple 2-matching ${\varvec{M}}_{\varvec{0}}$ of maximum cardinality and subsequently altering it into ${\varvec{M}}$ of the same cardinality so as to minimize the number of isolated edges. While the first phase (which computes ${\varvec{M}}_{\varvec{0}}$) runs in ${\varvec{O}}\left( {\varvec{E}} \sqrt{V}\right)$ time, the second one (which turns ${\varvec{M}}_{\varvec{0}}$ into ${\varvec{M}}$) requires ${\varvec{O(VE)}}$ time. In this paper we apply the general blocking augmentation approach (initially introduced, e.g., for bipartite matchings by Hopcroft and Karp, and also by Dinic) and present a novel algorithm that reduces the time needed for the second phase to ${\varvec{O}}\left( \textbf{E} \sqrt{V}\right)$ thus completely closing the gap between 1-restricted and unrestricted cases.