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 \(c_{1}, c_{2} :E \rightarrow \mathbb {R}\), we consider the optimization problem to find a basis \(X \in \mathcal {B}_{1}\) and a basis \(Y \in \mathcal {B}_{2}\) minimizing the 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 in Hradovich et al. (J Comb Optim 34(2):554–573, 2017). 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. We also present a strongly polynomial, primal-dual algorithm that computes a minimum cost solution for every feasible size of the intersection k in one run with asymptotic running time equal to one run of Frank’s matroid intersection algorithm. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids. We obtain a strongly polynomial time algorithm for the recoverable robust polymatroid base problem with interval uncertainties.

给定两个拟阵\(\mathcal {M}_{1} = (E, \mathcal {B}_{1})\)和\(\mathcal {M}_{2} = (E, \mathcal {B }_{2})\)在具有基集\(\mathcal {B}_1\)和\(\mathcal {B}_2\)的公共基础集E上，一些整数\(k \in \mathbb {N }\)和两个成本函数\(c_{1}, c_{2} :E \rightarrow \mathbb {R}\)，我们考虑优化问题以找到一个基\(X \in \mathcal {B} _{1}\)和一个基\(Y \in \mathcal {B}_{2}\)最小化成本\(\sum _{e\in X} c_1(e)+\sum _{e\在 Y} c_2(e)\)受下限约束\(|X \cap Y| \le k\), 上界约束\(|X \cap Y| \ge k\) ，或关于两个基X和Y的交集大小的等式约束\(|X \cap Y| = k\). 具有下界约束的问题原来是区间不确定性表示下的可恢复鲁棒拟阵问题的推广，对于该问题，强多项式时间算法的问题在 Hradovich 等人中作为一个悬而未决的问题。（J Comb Optim 34(2):554–573, 2017）。我们证明了对交集大小有上下界约束的两个问题可以简化为加权拟阵交集，从而用强多项式时间原始对偶算法来解决。我们还提出了一种强多项式原始对偶算法，该算法为交集k的每个可行大小计算最小成本解决方案在一次运行中，渐近运行时间等于 Frank 的拟阵交集算法的一次运行。此外，我们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概括。我们获得了具有区间不确定性的可恢复鲁棒多拟阵基问题的强多项式时间算法。