# Matroid Bases with Cardinality Constraints on the Intersection

## TL;DR

This paper develops strongly polynomial algorithms for optimizing matroid bases with intersection size constraints, including a novel solution for the equality constraint case, extending to polymatroids and multiple matroids.

## Contribution

It introduces a primal-dual algorithm for the matroid intersection problem with an equality constraint, solving an open problem and generalizing previous results.

## Key findings

- Polynomial-time algorithms for intersection constraints
- Solution for the equality constraint case
- Extensions to polymatroids and multiple matroids

## 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 $c_{1}, c_{2} \colon E \rightarrow \mathbb{R}$, 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.