In the Triangle-Free (Simple) 2-Matching problem we are given an undirected graph $G=(V,E)$. Our goal is to compute a maximum-cardinality $M\subseteq E$ satisfying the following properties: (1) at most two edges of $M$ are incident on each node (i.e., $M$ is a 2-matching) and (2) $M$ does not induce any triangle. In his Ph.D. thesis from 1984, Harvitgsen presents a complex polynomial-time algorithm for this problem, with a very complex analysis. This result was never published in a journal nor reproved in a different way, to the best of our knowledge. In this paper we have a fresh look at this problem and present a simple PTAS for it based on local search. Our PTAS exploits the fact that, as long as the current solution is far enough from the optimum, there exists a short augmenting trail (similar to the maximum matching case).

翻译：在三角形无关（简单）2匹配问题中，给定一个无向图 $G=(V,E)$。我们的目标是计算一个最大基数子集 $M\subseteq E$，满足以下性质：(1) 每个节点至多与 $M$ 中的两条边相关联（即 $M$ 是一个2匹配），且(2) $M$ 不诱导任何三角形。据我们所知，Harvitgsen 在1984年的博士论文中提出了针对该问题的一个复杂多项式时间算法，其分析过程非常繁复。该结果从未在期刊上发表，也未以其他方式被重新证明。本文重新审视该问题，并基于局部搜索提出一个简单的PTAS。我们的PTAS利用了如下事实：只要当前解距离最优解足够远，就存在一条短增广迹（类似于最大匹配情形）。