Efficient Matroid Intersection via a Batch-Update Auction Algorithm

TL;DR

This work presents a simple and generic auction algorithm that reduces $(1-\varepsilon)$-approximate matroid intersection to roughly $1/\varepsilon^2$ rounds of the easier problem of finding a maximum-weight basis of a single matroid.

Abstract

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same $n$-element ground set, the matroid intersection problem is to find a largest common independent set, whose size we denote by $r$. We present a simple and generic auction algorithm that reduces $(1-\varepsilon)$-approximate matroid intersection to roughly $1/\varepsilon^2$ rounds of the easier problem of finding a maximum-weight basis of a single matroid. Plugging in known primitives for this subproblem, we obtain both simpler and improved algorithms in two models of computation, including: * The first near-linear time/independence-query $(1-\varepsilon)$-approximation algorithm for matroid intersection. Our randomized algorithm uses $\tilde{O}(n/\varepsilon + r/\varepsilon^5)$ independence queries, improving upon the previous $\tilde{O}(n/\varepsilon + r\sqrt{r}/{\varepsilon^3})$ bound of Quanrud (2024). * The first sublinear exact parallel algorithms for weighted matroid intersection, using $O(n^{2/3})$ rounds of rank queries or $O(n^{5/6})$ rounds of independence queries. For the unweighted case, our results improve upon the previous $O(n^{3/4})$-round rank-query and $O(n^{7/8})$-round independence-query algorithms of Blikstad (2022).

Theorems & Definitions (32)

• theorem 1.1
• theorem 1.2
• remark 1.3
• theorem 1.4
• theorem 1.5
• lemma 2.1
• theorem 2.2: Matroid Intersection Theorem edmonds1979matroid
• claim 3.1
• proof
• lemma 3.2