A Weighted Kt, t-Free t-Factor Algorithm for Bipartite Graphs

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 Kt,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 Kt,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 restricted t-matchings.