Submodular Maximization in Exactly $n$ Queries

TL;DR

This paper introduces new deterministic algorithms for submodular maximization under matroid constraints that minimize the number of function queries, achieving optimal or near-optimal query complexity for both monotone and general cases.

Contribution

The paper presents the first algorithms with linear query complexity for submodular maximization under matroid constraints, including a 1/4 approximation with one query per element.

Findings

A 1/4 approximation algorithm for monotone submodular maximization using exactly one query per element.

A constant factor approximation algorithm for the general case with two queries per element.

First known algorithms with linear query complexity for this class of problems.

Abstract

In this work, we study the classical problem of maximizing a submodular function subject to a matroid constraint. We develop deterministic algorithms that are very parsimonious with respect to querying the submodular function, for both the case when the submodular function is monotone and the general submodular case. In particular, we present a 1/4 approximation algorithm for the monotone case that uses exactly one query per element, which gives the same total number of queries n as the number of queries required to compute the maximum singleton. For the general case, we present a constant factor approximation algorithm that requires 2 queries per element, which is the first algorithm for this problem with linear query complexity in the size of the ground set.