Deterministic $(2/3-\varepsilon)$-Approximation of Matroid Intersection Using Nearly-Linear Independence-Oracle Queries

TL;DR

This paper introduces a deterministic algorithm that approximates the maximum common independent set in two matroids with nearly linear oracle queries, improving efficiency and providing a semi-streaming variant.

Contribution

It presents the first deterministic $(2/3 - ext{epsilon})$-approximation algorithm for matroid intersection with nearly linear independence oracle queries.

Findings

Achieves a $(2/3 - ext{epsilon})$-approximation with $O(n/ ext{epsilon} + r ext{log} r)$ queries.

Utilizes a simple termination of a recent $(1 - ext{epsilon})$-approximation algorithm.

Provides a semi-streaming algorithm with $O(1/ ext{epsilon})$ passes.

Abstract

In the matroid intersection problem, we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ defined on the same ground set $V$ of $n$ elements, and the objective is to find a common independent set $S \in \mathcal{I}_1 \cap \mathcal{I}_2$ of largest possible cardinality, denoted by $r$. In this paper, we consider a deterministic matroid intersection algorithm with only a nearly linear number of independence oracle queries. Our contribution is to present a deterministic $O(\frac{n}{\varepsilon} + r \log r)$-independence-query $(2/3-\varepsilon)$-approximation algorithm for any $\varepsilon > 0$. Our idea is very simple: we apply a recent $\tilde{O}(n \sqrt{r}/\varepsilon)$-independence-query $(1 - \varepsilon)$-approximation algorithm of Blikstad [ICALP 2021], but terminate it before completion. Moreover, we also present a semi-streaming algorithm for $(2/3 -\varepsilon)$-approximation of matroid intersection in $O(1/\varepsilon)$ passes.