Submodular Maximization with Limited Function Access

TL;DR

This paper investigates submodular maximization with limited function access, proposing algorithms that use only pairwise evaluations, and introduces new curvature notions to analyze their performance and efficiency.

Contribution

It introduces algorithms for submodular maximization with pairwise information, establishes performance bounds, and defines new curvature measures, enhancing computational efficiency.

Findings

Algorithms outperform greedy in speed and approximation.

Performance bounds depend on submodular function curvature.

New curvature notions improve analysis of limited-access optimization.

Abstract

We consider a class of submodular maximization problems in which decision-makers have limited access to the objective function. We explore scenarios where the decision-maker can observe only pairwise information, i.e., can evaluate the objective function on sets of size two. We begin with a negative result that no algorithm using only $k$-wise information can guarantee performance better than $k/n$. We present two algorithms that utilize only pairwise information about the function and characterize their performance relative to the optimal, which depends on the curvature of the submodular function. Additionally, if the submodular function possess a property called supermodularity of conditioning, then we can provide a method to bound the performance based purely on pairwise information. The proposed algorithms offer significant computational speedups over a traditional greedy strategy. A by-product of our study is the introduction of two new notions of curvature, the $k$-Marginal Curvature and the $k$-Cardinality Curvature. Finally, we present experiments highlighting the performance of our proposed algorithms in terms of approximation and time complexity.