Subquadratic Submodular Maximization with a General Matroid Constraint

TL;DR

This paper introduces a faster randomized algorithm for monotone submodular maximization under a matroid constraint, achieving near-optimal approximation with subquadratic query complexity through a novel rounding technique.

Contribution

It presents a new subquadratic query complexity algorithm for submodular maximization with a general matroid constraint, featuring a novel rounding method using directed cycles.

Findings

Achieves a $(1 - 1/e - ext{ extepsilon})$-approximation with $ ilde{O}_{ ext{ extepsilon}}(\sqrt{r} n)$ queries.

Develops a new rounding algorithm using directed cycles of arbitrary length in an auxiliary graph.

Improves query complexity over previous algorithms by a significant factor.

Abstract

We consider fast algorithms for monotone submodular maximization with a general matroid constraint. We present a randomized $(1 - 1/e - \epsilon)$-approximation algorithm that requires $\tilde{O}_{\epsilon}(\sqrt{r} n)$ independence oracle and value oracle queries, where $n$ is the number of elements in the matroid and $r \leq n$ is the rank of the matroid. This improves upon the previously best algorithm by Buchbinder-Feldman-Schwartz [Mathematics of Operations Research 2017] that requires $\tilde{O}_{\epsilon}(r^2 + \sqrt{r}n)$ queries. Our algorithm is based on continuous relaxation, as with other submodular maximization algorithms in the literature. To achieve subquadratic query complexity, we develop a new rounding algorithm, which is our main technical contribution. The rounding algorithm takes as input a point represented as a convex combination of $t$ bases of a matroid and rounds it to an integral solution. Our rounding algorithm requires $\tilde{O}(r^{3/2} t)$ independence oracle queries, while the previously best rounding algorithm by Chekuri-Vondr\'{a}k-Zenklusen [FOCS 2010] requires $O(r^2 t)$ independence oracle queries. A key idea in our rounding algorithm is to use a directed cycle of arbitrary length in an auxiliary graph, while the algorithm of Chekuri-Vondr\'{a}k-Zenklusen focused on directed cycles of length two.