The weighted \({\mathcal {T}}\)-free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set \({\mathcal {T}}\) of triangles in G, find a maximum weight 2-matching containing no triangle in \({\mathcal {T}}\). When \({\mathcal {T}}\) is the set of all triangles in G, this problem is known as the weighted triangle-free 2-matching problem, which is a long-standing open problem. A main contribution of this paper is to give the first polynomial-time algorithm for the weighted \({\mathcal {T}}\)-free 2-matching problem under the assumption that \({\mathcal {T}}\) is a set of edge-disjoint triangles. In our algorithm, a key ingredient is to give an extended formulation representing the solution set, that is, we introduce new variables and represent the convex hull of the feasible solutions as a projection of another polytope in a higher dimensional space. Although our extended formulation has exponentially many inequalities, we show that the separation problem can be solved in polynomial time, which leads to a polynomial-time algorithm for the weighted \({\mathcal {T}}\)-free 2-matching problem.

无权重\（{\ mathcal {T}} \）的2个匹配问题是以下问题：给定无向图G，其边沿集上的权重函数和一个集\（{\ mathcal {T}} \）在三角形的ģ，查找包含在没有三角形的最大重量的2-匹配\（{\ mathcal【T}} \） 。当\（{\ mathcal {T}} \）是G中所有三角形的集合时，此问题称为无三角形加权2匹配问题，这是一个长期存在的开放问题。本文的主要贡献是在以下条件下给出了无权\（{\ mathcal {T}} \） 2匹配问题的第一个多项式时间算法\（{\ mathcal {T}} \）是一组不相交的三角形。在我们的算法中，关键因素是给出表示解集的扩展公式，即，我们引入新变量并将可行解的凸包表示为另一个多维空间在高维空间中的投影。尽管我们的扩展公式具有指数上的许多不等式，但我们证明了分离问题可以在多项式时间内解决，这导致了无加权\（{\ mathcal {T}} \） -2匹配问题的多项式时间算法。。