In this article, we introduce the b-bibranching problem in digraphs, which is a common generalization of the bibranching and b-branching problems. The bibranching problem, introduced by Schrijver, is a common generalization of the branching and bipartite edge cover problems. Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation. The b-branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa, is a generalization of the branching problem admitting higher indegree, that is, each vertex v can have indegree at most b(v). For b-branchings, a combinatorial algorithm, a linear programming formulation with total dual integrality, and a packing theorem for branchings are extended. A main contribution of this article is to extend those previous results on bibranchings and b-branchings to b-bibranchings. That is, we present a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for b-bibranchings. In particular, the linear program and M-convex submodular flow formulations, respectively, imply polynomial algorithms for finding a shortest b-bibranching.

中文翻译：

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b-双分支问题：TDI 系统、包装和离散凸性

在本文中，我们介绍了有向图中的b- bibranching 问题，它是 bibranching 和b- branching 问题的常见推广。Schrijver 引入的双分支问题是分支和二分边覆盖问题的常见推广。先前关于双分支的结果包括多项式算法、具有总对偶积分的线性规划公式、堆积定理和 M 凸子模流公式。最近由 Kakimura、Kamiyama 和 Takazawa 提出的b分支问题是对承认更高入度的分支问题的推广，即每个顶点v最多可以有b ( v ) 入度。为了b -分支、组合算法、具有完全对偶完整性的线性规划公式和分支的包装定理得到扩展。本文的一个主要贡献是将先前关于双分支和b -分支的结果扩展到b -双分支。也就是说，我们提出了一个具有总对偶积分的线性规划公式、一个堆积定理和一个用于b-双分支的 M-凸子模流公式。特别是，线性规划和 M-凸面子模流公式分别暗示了用于寻找最短b-双分支的多项式算法。