Matroid Intersection under Minimum Rank Oracle

TL;DR

This paper investigates the complexity of matroid intersection problems under a minimum rank oracle, presenting algorithms for specific cases, and proving NP-hardness and intractability in general and for polymatroids.

Contribution

It introduces a method to emulate augmenting path algorithms for unweighted matroid intersection under the oracle, and identifies special tractable cases and their limitations.

Findings

Constructed a necessary part of the exchangeability graph for unweighted case.

Proposed a fixed-parameter tractable algorithm based on circuit size.

Proved NP-hardness and intractability results for the general and polymatroid cases.

Abstract

In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the ranks of the given set in the two matroids. For the unweighted matroid intersection problem, we show how to construct a necessary part of the exchangeability graph, which enables us to emulate the standard augmenting path algorithm. For the weighted problem, the tractability is open in general. Nevertheless, we describe several special cases where tractability can be achieved, and we discuss potential approaches and the challenges encountered. On the positive side, we present a solution for the case where no circuit of one matroid is contained within a circuit of the other. Additionally, we propose a fixed-parameter tractable algorithm, parameterized by the maximum size of a circuit of one matroid. We also show that a lexicographically maximal common independent set can be found by the same approach, which leads to a nontrivial approximation ratio for finding a maximum-weight common independent set. On the negative side, we prove that the approach employed for the tractable cases above involves an NP-hard problem in the general case. We also show that if we consider the generalization to polymatroid intersection, even the unweighted problem is hard under the minimum rank oracle.