Separating Coverage and Submodular: Maximization Subject to a Cardinality Constraint

TL;DR

This work reveals a fundamental separation between Maximum Coverage and monotone Submodular Maximization under a cardinality constraint $k=cn$. It develops LP- and SDP-based rounding techniques to achieve a $c$-dependent approximation $\rho(c)$ for MC, while SM incurs a matching hardness near $\rho(c)$, demonstrating that coverage can outperform general monotone submodular objectives in this regime. A concrete breakthrough occurs at $c=1/2$, where MC achieves at least $0.7533$ while SM remains at $0.75$, establishing the first natural separation between these two well-studied problems. The paper also introduces improved MC algorithms for larger $c$ using SDP relaxations and a symmetry-gap-based hardness framework, and outlines a path to broader separations under plausible conjectures, with open questions about the optimal constants for both MC and SM and potential SDP-based improvements for all $c$. These results have implications for influence maximization, dataset summarization, and other applications where cardinality constraints interact with submodular structure, offering new algorithmic tools and hardness evidence for these foundational problems.

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.

Figures (1)

• Figure 1: $1-(1-c)^{1/c}$ (blue) is approximation for SM, $\rho(c)$ (green) is hardness for SM and approximation for MC, $\max\{1-1/e, c\}$ (dashed gray) is naive algorithm, and red point (proven) extended by dotted red line (numerical under a generalization of Austrin's simplicity conjecture A07, see \ref{['app:SDP']}) is approximation for MC. Both red dot and dotted red line separate MC and SM.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Pipage Rounding AS04CCPV07
• Theorem 2.2: Max $k$-Vertex-Cover ABG16
• Lemma 3.1
• proof
• proof : Proof of \ref{['thrm:LP-alg']}