Maximizing Non-Monotone Submodular Functions over Small Subsets: Beyond $1/2$-Approximation

TL;DR

This paper introduces two new algorithms for non-monotone submodular function maximization under a cardinality constraint, achieving better approximation ratios than previous methods, with one algorithm applicable offline and the other in streaming settings.

Contribution

It presents the first fixed parameter tractable algorithm with a 0.539-approximation and a streaming algorithm with a (1/2 + c)-approximation for symmetric functions, highlighting a separation for asymmetric cases.

Findings

Offline algorithm guarantees 0.539-approximation.

Streaming algorithm achieves (1/2 + c)-approximation for symmetric functions.

No space-efficient algorithm can surpass 1/2-approximation for asymmetric functions.

Abstract

In this work we give two new algorithms that use similar techniques for (non-monotone) submodular function maximization subject to a cardinality constraint. The first is an offline fixed parameter tractable algorithm that guarantees a $0.539$-approximation for all non-negative submodular functions. The second algorithm works in the random-order streaming model. It guarantees a $(1/2+c)$-approximation for symmetric functions, and we complement it by showing that no space-efficient algorithm can beat $1/2$ for asymmetric functions. To the best of our knowledge this is the first provable separation between symmetric and asymmetric submodular function maximization.