Subquadratic Weighted Matroid Intersection Under Rank Oracles

TL;DR

This paper introduces a new deterministic algorithm for weighted matroid intersection that significantly improves efficiency, achieving subquadratic query complexity and being the first to do so for polynomially-bounded weights.

Contribution

It presents the first subquadratic algorithm for weighted matroid intersection under rank oracles, improving previous methods with a simpler approach and better query complexity.

Findings

Achieves $ ilde{O}(nr^{3/4} ext{log}W)$ rank query complexity

First subquadratic algorithm for polynomially-bounded weights

Efficiently computes shortest-path trees in weighted exchange graphs

Abstract

Given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ over an $n$-element integer-weighted ground set $V$, the weighted matroid intersection problem aims to find a common independent set $S^{*} \in \mathcal{I}_1 \cap \mathcal{I}_2$ maximizing the weight of $S^{*}$. In this paper, we present a simple deterministic algorithm for weighted matroid intersection using $\tilde{O}(nr^{3/4}\log{W})$ rank queries, where $r$ is the size of the largest intersection of $\mathcal{M}_1$ and $\mathcal{M}_2$ and $W$ is the maximum weight. This improves upon the best previously known $\tilde{O}(nr\log{W})$ algorithm given by Lee, Sidford, and Wong [FOCS'15], and is the first subquadratic algorithm for polynomially-bounded weights under the standard independence or rank oracle models. The main contribution of this paper is an efficient algorithm that computes shortest-path trees in weighted exchange graphs.