A Min-Max Theorem for a Constrained Matching Problem

Abstract

The following constrained matching problem arises in the area of manpower scheduling. Consider an undirected graph $G=(V,E)$ and a digraph $D=(V,A)$. A master/slave-matching (MS-matching) in $G$ with respect to $D$ is a matching in $G$ such that for each arc $(u,v)\in A$ for which the node $u$ is matched, the node $v$ is matched too. The problem is to find an MS-matching of maximum cardinality. This paper addresses the special case where $G$ is bipartite with bipartition $V=W\cup U$ and every (weakly) connected component of $D$ is either an isolated node or two nodes in $U$ which are joined by a single arc. The polyhedral structure of this special case is investigated and a min-max theorem which characterizes the cardinality of a maximum MS-matching in terms of the weight of a special node cover is derived. This min-max theorem includes as a special case the theorem of Konig.