🤖 AI Summary
This paper studies the matroid intersection problem under the Minimum Rank Oracle (MRO) model. For the unweighted case, we reconstruct the exchange graph and rederive Edmonds’ min-max theorem solely via MRO queries, yielding the first polynomial-time algorithm. For the weighted case, we prove NP-hardness in general, but identify a tractable special case under circuit-disjointness and design an FPT algorithm parameterized by maximum circuit size. Furthermore, we develop a lexicographic optimization framework achieving a 1/2-approximation for maximum-weight common independent sets. Our key contributions are: (i) the first structural characterization of matroid intersection using only minimum-rank queries—bypassing reliance on standard rank or independence oracles; (ii) a novel paradigm for modeling exchange properties under MRO; and (iii) systematic results on parameterized tractability and approximation guarantees.
📝 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. Furthermore, we reformulate Edmonds' min-max theorem only using the minimum rank function, providing a new perspective on this result. 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. In particular, 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 circuit size. We also show that a lexicographically maximal common independent set can be found by the same approach, which leads to at least $1/2$-approximation for finding a maximum-weight common independent set.