A Unified Approach to Submodular Maximization Under Noise

TL;DR

This paper introduces a meta-algorithm that transforms existing exact submodular maximization algorithms into robust solutions capable of handling noisy oracle access, achieving near-optimal approximation guarantees under noise.

Contribution

The authors develop a universal meta-algorithm that preserves approximation ratios for submodular maximization problems in noisy settings, applicable to various constraints and algorithms.

Findings

Achieves a (1-1/e)-approximation for monotone submodular maximization under noise.

Attains a 1/e-approximation for non-monotone submodular maximization with noise.

Provides a 1/2-approximation for unconstrained non-monotone submodular maximization under noise.

Abstract

We consider the problem of maximizing a submodular function with access to a noisy value oracle for the function instead of an exact value oracle. Similar to prior work, we assume that the noisy oracle is persistent in that multiple calls to the oracle for a specific set always return the same value. In this model, Hassidim and Singer (2017) design a $(1-1/e)$-approximation algorithm for monotone submodular maximization subject to a cardinality constraint, and Huang et al (2022) design a $(1-1/e)/2$-approximation algorithm for monotone submodular maximization subject to any arbitrary matroid constraint. In this paper, we design a meta-algorithm that allows us to take any "robust" algorithm for exact submodular maximization as a black box and transform it into an algorithm for the noisy setting while retaining the approximation guarantee. By using the meta-algorithm with the measured continuous greedy algorithm, we obtain a $(1-1/e)$-approximation (resp. $1/e$-approximation) for monotone (resp. non-monotone) submodular maximization subject to a matroid constraint under noise. Furthermore, by using the meta-algorithm with the double greedy algorithm, we obtain a $1/2$-approximation for unconstrained (non-monotone) submodular maximization under noise.