Submodular Maximization in Exactly $n$ Queries

TL;DR

This work tackles the problem of submodular maximization under matroid constraints, presenting deterministic algorithms with minimal query overhead. It introduces QuickSwap, an $n$-query $1/4$-approximation for MSM, and extends the framework to GSM with a linear-query $2n$-query constant-factor algorithm, as well as to MSPM under $p$-matchoids achieving $1/(4p)$. The methods hinge on swap-based exchanges, maintaining an infeasible supersolution to drive decisions, and a graph-based analysis that bounds performance via injections from the optimum. Empirically, QuickSwap demonstrates substantial gains in query efficiency while preserving competitive objective values on influence-maximization-like tasks under partition matroids. These results collectively advance the practical efficiency of submodular maximization in large-scale settings, and invite future work on tighter ratios and broader constraint families.

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.

Figures (1)

• Figure 1: The numbers of queries and objective values achieves by the four algorithms for MSM as a function of rank of the matroid. For the objective value plots, lazy greedy and threshold greedy achieve nearly identical objective values and their lines overlap. For the same reason, the lines for QuickSwap and CK also overlap for these plots.

Theorems & Definitions (45)

• Theorem
• Theorem
• Theorem
• lemma 2.1
• lemma 2.2
• theorem 2.1
• proof
• lemma 2.3
• proof
• proof : Proof of Lemma \ref{['lem:primehalf']}