The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis.

中文翻译：

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最短二分支的两种公式的关系

最短双分支问题是二部图中最小权重边覆盖问题和有向图中最小权重树状问题的普遍推广。对于最短的双分支问题，Keijsper 和 Pendavingh (1998) 给出了一种有效的原始对偶算法，并且 Schrijver (1982) 将问题的易处理性归因于线性规划公式中的总对偶积分。Takazawa (2012) 的估计拟阵交集公式提供了关于该问题易处理性的另一种观点。在本文中，我们讨论了这两种最短分支问题的公式之间的关系。我们首先证明了可通过 Benders 分解从线性规划公式推导出已赋值拟阵相交公式，其中在分解过程中保留了完整性，并且由此产生的凸规划被赋予了离散凸性。然后，我们展示了一个公式的一对原始和对偶最优解是如何从另一个公式构建的，从而提供多面体组合学和离散凸分析之间的联系。