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A Note On Deterministic Submodular Maximization With Bounded Curvature

Wenxin Li

TL;DR

This work discusses deterministic maximization of a monotone submodular function under a matroid constraint when the function has bounded curvature $κ_f$. It builds on the breakthrough deterministic algorithm of Buchbinder and Feldman by showing that, using (approximate) local maxima within a non-oblivious local search guided by a constructed potential $Φ_g$, one can retain the $(1 - κ_f/e - ε)$-approximation. The section introduces the curvature $κ_f$, the submodularity ratio $γ_f$ (and $γ_g$ for a related function), and a key lemma involving the potential $Φ_g$. The results unify ideas from recent deterministic submodular maximization literature and provide a pathway to deterministic guarantees for functions with bounded curvature under matroid constraints.

Abstract

We show that the recent breakthrough result of [Buchbinder and Feldman, FOCS'24] could further lead to a deterministic $(1-κ_{f}/e-\varepsilon)$-approximate algorithm for maximizing a submodular function with curvature $κ_{f}$ under matroid constraint.

A Note On Deterministic Submodular Maximization With Bounded Curvature

TL;DR

This work discusses deterministic maximization of a monotone submodular function under a matroid constraint when the function has bounded curvature . It builds on the breakthrough deterministic algorithm of Buchbinder and Feldman by showing that, using (approximate) local maxima within a non-oblivious local search guided by a constructed potential , one can retain the -approximation. The section introduces the curvature , the submodularity ratio (and for a related function), and a key lemma involving the potential . The results unify ideas from recent deterministic submodular maximization literature and provide a pathway to deterministic guarantees for functions with bounded curvature under matroid constraints.

Abstract

We show that the recent breakthrough result of [Buchbinder and Feldman, FOCS'24] could further lead to a deterministic -approximate algorithm for maximizing a submodular function with curvature under matroid constraint.
Paper Structure (1 section, 3 theorems, 16 equations)

This paper contains 1 section, 3 theorems, 16 equations.

Table of Contents

  1. Guessing the submodular ratio.

Key Result

Theorem 1

There exists a deterministic polynomial time algorithm that achieves an approximation ratio of $(1-\kappa_{f}/e-\varepsilon)$ for maximizing a submodular function $f(\cdot)$ with bounded curvature $\kappa_{f}$ under the matroid constraint.

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1: Curvature sviridenko2017optimal
  • Definition 2: Submodular Ratio
  • Lemma 1: buchbinder2024deterministic
  • proof
  • Lemma 2
  • proof