Triangle-free 2-matchings

TL;DR

This work addresses the problem of computing a maximum triangle-free $2$-matching in a graph by introducing a substantially simpler algorithm that runs in $O(|V||E|)$ time. Central to the approach is restricting augmentations to amenable $M$-augmenting paths and employing a decomposition framework that expresses the symmetric difference of two triangle-free $2$-matchings as amenable $M$-alternating paths and cycles. The method relies on a dynamic structure $S$ and a transformed graph $G'$ with half-edges to manage reachability and edge-forbidding, along with a precise handling of dense subgraphs and blossoms. The main contributions are a decomposition theorem for triangle-free $2$-matchings, a practical $O(nm)$ algorithm to obtain maximum solutions, and a careful analysis that avoids the complexity of prior approaches, with potential implications for related matching and routing problems.

Abstract

We consider the problem of finding a maximum size triangle-free $2$-matching in a graph $G=(V,E)$. A (simple) $2$-matching is any subset of the edges such that each vertex is incident to at most two edges from the subset. The first polynomial time algorithm for this problem was given by Hartvigsen in 1984 in his PhD thesis and its improved version has been recently published in a journal. We present a different, significantly simpler algorithm with a relatively short proof of correctness. Our algorithm with running time $O(|V||E|)$ is additionally faster than the one by Hartvigsen having running time $O(|V|^3|E|^2)$. It has been proven before that for any triangle-free $2$-matching $M$ which is not maximum the graph contains an $M$-augmenting path, whose application to $M$ results in a bigger triangle-free $2$-matching. A new observation is that the search for an augmenting path $P$ can be restricted to so-called {\em amenable} paths that go through any triangle $t$ contained in $P\cup M$ a limited number of times. Amenable paths can be characterised with the aid of {\em half-edges}. A {\em half-edge} of edge $e$ is, informally speaking, a half of $e$ containing exactly one of its endpoints. Each half-edge serves also as a {\em hinge} - a connector between one pair of edges on an alternating path. To find an amenable augmenting path we thus dynamically remove and re-add half-edges to forbid or allow some edges to be followed by certain others. The existence of amenable augmenting paths follows from our decomposition theorem for triangle-free $2$-matchings. This decomposition theorem is largely the same as the decomposition from versions 1-6 of this paper and is moreover simpler and stronger than the one given by Kobayashi and Noguchi.

Figures (16)

• Figure 1: An amenable path $(s,a,b,c,a,d,f,z)$.
• Figure 2: A bumpy path $(a,b,c)$.
• Figure 3: Dense subgraphs on $a,b,c,d,x$.
• Figure 4: In each case the application to $M$ of an augmenting path from $s$ to $t$ creates a triangle $t=(a,b,c)$.
• Figure 5: At the point when we have not checked adding the edge $(g,a)$ yet (it might also be the point when we have already added the edge $(a,g)$ but haven't added $(f,g)$ or $(f,b)$), the triangle $t=(a,b,c)$ is vulnerable and we remove the hinge $h(b,c,e)=(x^c_{cb},c_2)$ from $G_2$. After the addition of $(a,g)$ and $(a,b)$ to $S$ the triangle $t$ ceases to be vulnerable, because $S$ contains a path $(s,d,b,f,g,a,b,c,e)$ that is amenable on $t$ and hence $h(b,c,e)$ is passable and we restore it in $G_2$.

Theorems & Definitions (43)

• Corollary 1
• Theorem 1
• proof
• Claim 1
• Corollary 2
• proof
• Definition 1
• Definition 2
• Lemma 1
• Lemma 2