The Weighted Spanning Tree Constraint Revisited

The weighted spanning tree constraint, or wst-constraint, is defined on an edgeweighted graph G and a value K. It states that G admits a spanning tree with weight at most K [3, 4]. It can be applied to network design problems as well as routing problems, in which it serves as a relaxation. In this work, we assume that we can represent the mandatory and possible edges that can belong to a solution to the wst-constraint, e.g., using a subset-bound set variable as in [3]. Dooms and Katriel [3] consider a version of the wst-constraint in which the weights of the edges are also variable. They propose several filtering algorithms, including one for the version of the wst-constraint that we consider in this paper. Subsequently, a more practical and incremental filtering algorithm for this constraint was proposed by Régin [4]. In this work, we extend the algorithm of Régin [4] in several ways. First, we revisit the computation of the ‘replacement cost’ of tree edges, and present an algorithm with an almost linear time complexity. Second, we take mandatory edges into account; that is, edges that belong to every spanning tree having a weight at most K or that are imposed by the user. Third, we discuss the incremental behavior of the algorithms when mandatory edges are introduced.