In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds of finite 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 (2013) [9].

中文翻译：

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通过广义分数拟阵匹配最小化钻石上的子模函数

在本文中，我们展示了第一个多项式时间算法，用于最小化有限尺寸钻石乘积上的子模函数。这个子模函数最小化问题被简化为关联多面体的隶属问题，相当于基于椭球方法的多面体上的优化问题。后一个优化问题是加权分数拟阵匹配问题的推广。我们通过扩展 Gijswijt 和 Pap (2013) [9] 的结果，为这个优化问题提供了一个组合多项式时间算法。