This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Miller, Owen and Provan's algorithm (2015) as a subroutine. Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.

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For a given nonnegative matrix A = (A ij ), the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix XAY for some positive diagonal matrices X, Y . The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization A ij ← A ij / j A ij and column-normalization A ij ← A ij / i A ij 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, Samorodnitsky, and Wigderson showed that a polynomial number of the Sinkhorn iterations for A G decides whether G has a perfect matching.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 certificate of the nonexistence of a perfect matching. Our analysis is based on an interpretation of the Sinkhorn algorithm as alternating KL-divergence minimization (Csiszár and Tusnády 1984, Gietl andReffel 2013) and its limiting behavior for a nonscalable matrix (Aas 2014). We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage-Mendelsohn decomposition of a bipartite graph.

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