交集上具有基数约束的拟阵基
Matroid Bases with Cardinality Constraints on the Intersection
摘要：
给定两个拟阵$\mathcal{M}_{1} = (E, \mathcal{B}_{1})$和$\mathcal{M}_{2} = (E, \mathcal{B}_{2})$在一个公共基集$E$上，具有基集$\mathcal{B}_{1}$和$\mathcal{B}_{2}$，某个整数$k \in \mathbb{N}$，以及两个代价函数$c_{1}, c_{2} \colon E \rightarrow \mathbb{R}$，我们考虑优化问题，以找到一个基$X \in \mathcal{B}_{1}$和一个基$Y \in \mathcal{B}_{2}$，使代价$\sum_{e\in X} c_1(e)+\sum_{e\in Y} c_2(e)$受到下界约束$|X \cap Y| \le k$，上界约束$|X \cap Y| \ge k$，或者对两个基$X$和$Y$的交集的大小的等式约束$|X \cap Y| = k$。具有下界约束的问题是区间不确定性表示下的可恢复鲁棒拟阵问题的推广，Hradovich等人将其强多项式时间算法问题作为一个开放问题。我们证明了这两个具有下界和上界约束的问题可以归结为加权拟阵交问题，因此可以用强多项式时间原始对偶算法来求解。等式约束的问题是否也能有效地解决，这一问题变得更加困难。作为我们的主要结果，我们提出了一个强多项式，原始-对偶算法的等式约束的问题的交集的大小。此外，我们还讨论了从拟阵到多拟阵，以及从两个到三个或更多拟阵的问题的推广。
英文摘要：
Given two matroids $\mathcal{M}_{1} = (E, \mathcal{B}_{1})$ and $\mathcal{M}_{2} = (E, \mathcal{B}_{2})$ on a common ground set $E$ with base sets $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, some integer $k \in \mathbb{N}$, and two cost functions $c_{1}, c_{2} \colon E \rightarrow \mathbb{R}$, we consider the optimization problem to find a basis $X \in \mathcal{B}_{1}$ and a basis $Y \in \mathcal{B}_{2}$ minimizing cost $\sum_{e\in X} c_1(e)+\sum_{e\in Y} c_2(e)$ subject to either a lower bound constraint $|X \cap Y| \le k$, an upper bound constraint $|X \cap Y| \ge k$, or an equality constraint $|X \cap Y| = k$ on the size of the intersection of the two bases $X$ and $Y$. The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomial-time algorithm was left as an open question by Hradovich et al. We show that the two problems with lower and upper bound constraints on the size of the intersection can be reduced to weighted matroid intersection, and thus be solved with a strongly polynomial-time primal-dual algorithm. The question whether the problem with equality constraint can also be solved efficiently turned out to be a lot harder. As our main result, we present a strongly-polynomial, primal-dual algorithm for the problem with equality constraint on the size of the intersection. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids.
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