A weighted K t,t -free t-factor algorithm for bipartite graphs

Abstract

For a simple bipartite graph and an integer t≥2, we consider the problem of finding a minimum-weight t-factor under the restriction that it contains no complete bipartite graph K t,t as a subgraph. When t∈=∈2, this problem amounts to the minimum-weight square-free 2-factor problem in a bipartite graph, which is NP-hard. We propose, however, a strongly polynomial algorithm for a certain case where the weight vector is vertex-induced on any subgraph isomorphic to K t,t . The algorithm adapts the unweighted algorithms of Hartvigsen and Pap, and a primal-dual approach to the minimum-cost flow problem. The algorithm is fully combinatorial, and thus provides a dual integrality theorem, which is tantamount to Makai's theorem dealing with maximum-weight K t,t -free t-matchings. © 2008 Springer-Verlag Berlin Heidelberg.