Minimizing Submodular Functions on Diamonds via Generalized Fractional Matroid Matchings

In this paper we show the rst polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds of nite size. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap [D. Gijswijt and G. Pap, An algorithm for weighted fractional matroid matching, J. Combin. Theory, Ser. B 103 (2013), 509 520].