Finding Hall Blockers by Matrix Scaling

Given a nonnegative matrix A = ( A i j ) , the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix D 1 A D 2 for some positive diagonal matrices D 1 , D 2 . The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization A i j ← A i j / ∑ j A i j and column-normalization A i j ← A i j / ∑ i A i j 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 A G . Linial et al. (2000) showed that O ( n 2 log n ) iterations for A G 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 Γ ( X ) with | X | > | Γ ( X ) | , which is a certificate of the nonexistence of a perfect matching. Specifically, we show that O ( n 2 log n ) iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing ( 1 − λ ) | X | − λ | Γ ( X ) | for λ ∈ [ 0 , 1 ] . 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.