Finding Hall Blockers by Matrix Scaling

Abstract

Given a nonnegative matrix [Formula: see text], the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix [Formula: see text] for some positive diagonal matrices D1, D2. The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization [Formula: see text] and column-normalization [Formula: see text] alternatively. By this algorithm, A converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with A has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph G, which is identified with the 0, 1-matrix AG. Linial et al. (2000) showed that [Formula: see text] iterations for AG decide whether G has a perfect matching. Here, n is the number of vertices in one of the color classes of G. In this paper, we show an extension of this result. If G has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker—a vertex subset X having neighbors [Formula: see text] with [Formula: see text], which is a certificate of the nonexistence of a perfect matching. Specifically, we show that [Formula: see text] iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing [Formula: see text] for [Formula: see text]. The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for Kullback–Leibler (KL) divergence and on its limiting behavior for a nonscalable matrix. We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage–Mendelsohn decomposition of a bipartite graph. Funding: K. Hayashi was supported by the Japan Society for the Promotion of Science [Grant JP19J22605]. H. Hirai was supported by Precursory Research for Embryonic Science and Technology [Grant JPMJPR192A].