A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to determine an optimal connected matching in an edge-weighted graph, even in the planar bipartite case. We present two mixed integer programming formulations and a sophisticated branch-and-cut scheme to find weighted connected matchings in general graphs. The formulations explore different polyhedra associated to this problem, including strong valid inequalities both from the matching polytope and from the connected subgraph polytope. We conjecture that one attains a tight approximation of the convex hull of connected matchings using our strongest formulation, and report encouraging computational results over DIMACS Implementation Challenge benchmark instances. The source code of the complete implementation is also made available.

翻译：在图G中，连通匹配由一组两两不相交的边组成，这些边覆盖的顶点在G中诱导出一个连通子图。虽然寻找最大基数连通匹配是一个已得到良好解决的问题，但在边加权图中（即使是在平面二分图情形下）确定最优连通匹配是NP难的。我们提出了两种混合整数规划公式和一个复杂的分支切割方案，用于在一般图中寻找加权连通匹配。这些公式探索了与该问题相关的不同多面体，包括来自匹配多面体和连通子图多面体的强有效不等式。我们猜想，使用我们最强的公式能够实现对连通匹配凸包的紧逼近，并在DIMACS实施挑战基准实例上报告了令人鼓舞的计算结果。完整实现的源代码也已公开提供。