Matroid Bases with Cardinality Constraints on the Intersection

Given two matroids M ₁= (E, B ₁) and M ₂= (E, B ₂) on a common ground set E with base sets B ₁ and B ₂, some integer k∈ N, and two cost functions c ₁, c ₂: E→ R, we consider the optimization problem to find a basis X∈ B ₁ and a basis Y∈ B ₂ minimizing the cost ∑ _{e ∈ X}c ₁(e) + ∑ _{e ∈ Y}c ₂(e) subject to either a lower bound constraint | X∩ Y| ≤ k, an upper bound constraint | X∩ Y| ≥ k, or an equality constraint | X∩ 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.