Separating Coverage and Submodular: Maximization Subject to a Cardinality Constraint

2024-11-08

arXiv.org

AI Summary

This paper investigates the fundamental gap in approximation performance between the maximum coverage problem and general monotone submodular maximization under a cardinality constraint equal to a fixed fraction $c$ of the ground set size. Contrary to the conventional belief that both problems share identical approximation limits, we establish the first provable separation: when $c = 1/2$, maximum coverage admits a $0.7533$-approximation—strictly exceeding the best possible approximation ratio for arbitrary monotone submodular functions. Technically, we derive a tight approximation bound $1 - (1 - c)^{1/c}$ for the submodular case and provide a matching hardness lower bound, particularly tight when $c = 1/s$ for integer $s$. Our analysis integrates constructive hardness proofs, an improved greedy algorithm, and refined perturbation arguments. These results collectively demonstrate an intrinsic divergence in the optimal approximation ratios of the two problems under proportional cardinality constraints.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Comparing approximation guarantees for coverage vs. submodular maximization under cardinality constraints.
• Deriving new approximation bounds for submodular maximization with constant fraction constraints.
• Demonstrating separation in approximation ratios between coverage and submodular objectives.

Innovation

• Greedy algorithm for submodular maximization
• Approximation ratio varies with cardinality
• Different approximations for coverage vs submodular