Consistency of the Matching Predicate

Abstract

Let G(V,E) denote an undirected graph, V and E being the sets of its nodes and edges, respectively. A matching in G(V,E) is a subset of edges with no common endpoints. Finding a matching of maximum cardinality constitutes the maximum cardinality matching (MCM) problem. For a thorough theoretical discussion we refer to [6]. The MCM problem is of specific interest from a Constraint Programming (CP) point of view because it can model several logical constraints (predicates) like the all_different and the symmetric all_different predicates [7]. Thus, the definition of a maximum cardinality matching constraint provides a framework encompassing other predicates. Along this line of research, we define a global constraint with respect to the MCM and address the issue of consistency. Establishing hyper-arc consistency implies the identification of edges that cannot participate in any maximum cardinality matching. Evidently, this issue (also called filtering) is related to the methods developed for solving the problem. Solving this problem for bipartite graphs was common knowledge long before Edmonds proposed an algorithm for the non-bipartite case [3]. Regarding hyper-arc consistency, the problem has been resolved only for the bipartite case [1].