Robust Submodular Maximization: A Non-Uniform Partitioning Approach

TL;DR

This paper introduces a new algorithm for robust submodular maximization that extends approximation guarantees to larger removal sizes, improving performance in data summarization and influence maximization tasks.

Contribution

It presents the Partitioned Robust (PRo) algorithm that achieves approximation guarantees for larger removal fractions, solving an open problem in robust submodular maximization.

Findings

PRo outperforms greedy algorithms in experiments.

Demonstrates improved robustness in data summarization.

Shows effectiveness in influence maximization tasks.

Abstract

We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $\tau$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $\tau = o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $\tau = o(k)$. Our algorithm constructs partitions consisting of buckets with exponentially increasing sizes, and applies standard submodular optimization subroutines on the buckets in order to construct the robust solution. We numerically demonstrate the performance of PRo in data summarization and influence maximization, demonstrating gains over both the greedy algorithm and the algorithm of (Orlin et al., 2016).