Matroid Bases with Cardinality Constraints on the Intersection

Abstract

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 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 polynomialtime algorithm was left as an open question in [6].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.

An Investigation of the Recoverable Robust Assignment Problem

“…Hradovic, Kasperski & Zieliński [11,10] obtain a polynomial time algorithm for the recoverable matroid basis problem and a strongly polynomial time algorithm for the recoverable spanning tree problem. These results have been generalized and improved by Lendl, Peis & Timmermans [15] who show that the recoverable matroid basis and the recoverable polymatroid basis problem can both be solved in strongly polynomial time. Iwamasa & Takayawa [12] further generalize these results and cover cases with nonlinear and convex cost functions.…”

Recoverable Robust Representatives Selection Problems with Discrete Budgeted Uncertainty

“…This is a special case of the recoveralbe robust matroid basis problem with interval uncertainty studied in [LPT19], where a strongly polynomial time algorithm for this problem is given. Our problem corresponds to the special case of partition matroids, hence, we arrive at the following observation.…”

Optimal matroid bases with intersection constraints: Valuated matroids, M-convex functions, and their applications

“…The motivation of these problems comes from the recoverable robust matroid basis problem [2]. Lendl et al [17] showed that the problems (1)-( 3) are polynomial-time solvable: they developed a new primal-dual algorithm for the problem (1); and reduced the problems ( 2) and (3) to weighted matroid intersection. By this result, they affirmatively settled an open question on the strongly polynomial-time solvability of the recoverable robust matroid basis problem under interval uncertainty representation [11,12].…”

“…Lendl et al [17] further discussed two kinds of generalizations of the above problems. One is a polymatroidal generalization.…”

“…Recently, Lendl, Peis, and Timmermans [17] have introduced the following variants of weighted matroid intersection, in which a cardinality constraint is imposed on the intersection. Let V be a finite set, and M 1 = (V, B 1 ) and M 2 = (V, B 2 ) be matroids on V with base families B 1 and B 2 , respectively.…”