A New Approximation Guarantee for Monotone Submodular Function Maximization via Discrete Convexity

TL;DR

This paper introduces a novel approximation guarantee for monotone submodular maximization by leveraging discrete convex analysis, specifically through extracting an M$^ atural$-concave function and defining $h$-curvature, leading to improved approximation bounds.

Contribution

The paper proposes a new approximation guarantee based on $h$-curvature, improving upon traditional curvature-based bounds for submodular maximization problems.

Findings

Achieves a $(1-rac{ ext{h-curvature}}{e}- ext{epsilon})$-approximation in polynomial time.

Introduces the concept of $h$-curvature to measure deviation from M$^ atural$-concavity.

Provides nontrivial approximation guarantees for various submodular maximization problems.

Abstract

In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature. However, the notion of curvature is often pessimistic, and we rarely obtain improved approximation guarantees, even for very simple objective functions. In this paper, we provide a novel approximation guarantee by extracting an M$^\natural$-concave function $h:2^E \to \mathbb R_+$, a notion in discrete convex analysis, from the objective function $f:2^E \to \mathbb R_+$. We introduce the notion of $h$-curvature, which measures how much $f$ deviates from $h$, and show that we can obtain a $(1-\gamma/e-\epsilon)$-approximation to the problem of maximizing $f$ under a cardinality constraint in polynomial time for any constant $\epsilon > 0$. Then, we show that we can obtain nontrivial approximation guarantees for various problems by applying the proposed algorithm.