Separating Coverage and Submodular: Maximization Subject to a Cardinality Constraint

TL;DR

This paper investigates the approximation limits of maximum coverage and monotone submodular maximization under a fractional cardinality constraint, revealing a separation in their best achievable approximation ratios.

Contribution

It introduces a new approximation ratio for monotone submodular maximization with fractional constraints and demonstrates a separation from maximum coverage, which was previously assumed to be similar.

Findings

Monotone submodular maximization admits an approximation of 1-(1-c)^{1/c}.

Maximum Coverage with c=1/2 has an approximation of 0.7533, separating it from submodular maximization.

The paper establishes matching hardness results for certain fractional constraints.

Abstract

We consider two classic problems: maximum coverage and monotone submodular maximization subject to a cardinality constraint. [Nemhauser--Wolsey--Fisher '78] proved that the greedy algorithm provides an approximation of $1-1/e$ for both problems, and it is known that this guarantee is tight ([Nemhauser--Wolsey '78; Feige '98]). Thus, one would naturally assume that everything is resolved when considering the approximation guarantees of these two problems, as both exhibit the same tight approximation and hardness. In this work we show that this is not the case, and study both problems when the cardinality constraint is a constant fraction $c \in (0,1]$ of the ground set. We prove that monotone submodular maximization subject to a cardinality constraint admits an approximation of $1-(1-c)^{1/c}$; This approximation equals $1$ when $c=1$ and it gracefully degrades to $1-1/e$ when $c$ approaches $0$. Moreover, for every $c=1/s$ (for any integer $s \in \mathbb{N}$) we present a matching hardness. Surprisingly, for $c=1/2$ we prove that Maximum Coverage admits an approximation of $0.7533$, thus separating the two problems. To the best of our knowledge, this is the first known example of a well-studied maximization problem for which coverage and monotone submodular objectives exhibit a different best possible approximation.