公平积分次模流
Fair Integral Submodular Flows
摘要：
研究了整数次模流多面体$Q$的整数值元素，它们是递减最小的（dec-min），即它们的最大分量尽可能小，在此范围内，第二大分量尽可能小，以此类推。我们证明了dec-$Q$的min整元是另一个积分子模流多面体的整元的集合，该积分子模流多面体通过与以下面相交而从$Q$产生：带盒子的$Q$。基于这种描述，我们开发了一个强多项式算法计算不仅是一个dec-min整数值子模流，但即使是最便宜的一个线性成本函数。一个特殊的情况是寻找一个混合图的强连通（或$k$ -边连通）定向，其入度向量是递减最小的。
英文摘要：
Integer-valued elements of an integral submodular flow polyhedron $Q$ are investigated which are decreasingly minimal (dec-min) in the sense that their largest component is as small as possible, within this, the second largest component is as small as possible, and so on. As a main result, we prove that the set of dec-min integral elements of $Q$ is the set of integral elements of another integral submodular flow polyhedron arising from $Q$ by intersecting a face of $Q$ with a box. Based on this description, we develop a strongly polynomial algorithm for computing not only a dec-min integer-valued submodular flow but even a cheapest one with respect to a linear cost-function. A special case is the problem of finding a strongly connected (or $k$-edge-connected) orientation of a mixed graph whose in-degree vector is decreasingly minimal.
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