Deterministic $(2/3-\varepsilon)$-Approximation of Matroid Intersection Using Nearly-Linear Independence-Oracle Queries
Tatsuya Terao
TL;DR
The paper addresses the matroid intersection problem for two matroids on a common ground set $V$ with maximum common independent set size $r$, aiming for near-linear independence‑query efficiency. The authors design a deterministic $2/3-ε$‑approximation algorithm that leverages nearly linear independence‑queries by terminating Blikstad's $1-ε$‑approximation algorithm early, achieving a query complexity of $O(n/ε + r \log r)$ and a runtime tied to the independence oracle cost. A rank‑oracle variant achieves the same approximation with $O(n/ε)$ rank queries, and a semi‑streaming implementation delivers a $2/3-ε$‑approximation in $O(1/ε)$ passes with memory suitable for $(r_1+r_2)$. Overall, the work advances deterministic matroid intersection methods under near‑linear query regimes and connects independence, rank queries, and streaming models.
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.