Matroid Intersection under Restricted Oracles

TL;DR

This paper investigates the complexity of matroid intersection problems when using restricted oracles, providing new polynomial-time algorithms and tractability results for specific oracle models and matroid classes.

Contribution

It introduces a strongly polynomial algorithm for weighted matroid intersection with the rank sum oracle and establishes tractability for unweighted intersection with partition matroids and weighted cases with elementary split matroids.

Findings

Polynomial algorithm for weighted intersection under rank sum oracle

Tractability of unweighted intersection with partition matroids

Weighted intersection solvable with elementary split matroids

Abstract

Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the rank sum, common independence, and maximum rank oracles. We give a strongly polynomial-time algorithm for weighted matroid intersection under the rank sum oracle. In the common independence oracle model, we prove that the unweighted matroid intersection problem is tractable when one of the matroids is a partition matroid, and that even the weighted case is solvable when one of the matroids is an elementary split matroid. Finally, we show that the common independence and maximum rank oracles together are strong enough to realize the steps of our algorithm under the rank sum oracle.