Matroid bases with cardinality constraints on the intersection

Stefan Lendl,Veerle Timmermans,Britta Peis

doi:10.1007/s10107-021-01642-1

Abstract

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 _{ein X} c_1(e)+sum _{ein 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.

Highlights

Matroids are fundamental and well-studied structures in combinatorial optimization
We saw in the previous section that both problems, (P≤k) and (P≥k), can be solved in strongly polynomial time via a weighted matroid intersection algorithm
Remark 1 Note that an alternative method to solve (P=k) for a given fixed k is to compute solutions (X1, Y1), (X2, Y2) for both (P≤k) and (P≥k) respectively, which can be done by running a matroid intersection algorithm twice

Summary

Introduction

Matroids are fundamental and well-studied structures in combinatorial optimization. Recall that a matroid M is a tuple M = (E, F), consisting of a finite ground set E and a family of subsets F ⊆ 2E , called the independent sets, satisfying (i) ∅ ∈ F, (ii) if F ∈ F and F ⊂ F, F ∈ F, and (iii) if F, F ∈ F with |F | > |F|, there exists some element e ∈ F \ F satisfying F ∪ {e} ∈ F. The problem to find a min-cost common base in two matroids M1 = (E, B1) and M2 = (E, B2) on the same ground set, or the problem to maximize a linear function over the intersection F1 ∩ F2 of two matroids M1 = (E, F1) and M2 = (E, F2) can be done efficiently with a strongly-polynomial primal-dual algorithm (cf [5]). We saw in the previous section that both problems, (P≤k) and (P≥k), can be solved in strongly polynomial time via a weighted matroid intersection algorithm This leads to the question whether we can solve the problem (P=k) with equality constraint on the size of the intersection efficiently as well, and whether a parametric algorithm exists, which computes the whole parametric curve with respect to k

The algorithm

A generalization to polymatroid base polytopes

More than two matroids

B Proof of Theorem 2