Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

Shengminjie Chen, Yiwei Gao, Kaifeng Lin, Xiaoming Sun, Jialin Zhang

Abstract

Submodular maximization constitutes a prominent research topic in combinatorial optimization and theoretical computer science, with extensive applications across diverse domains. While substantial advancements have been achieved in approximation algorithms for submodular maximization, the majority of algorithms yielding high approximation guarantees are randomized. In this work, we investigate deterministic approximation algorithms for maximizing non-monotone submodular functions subject to matroid and knapsack constraints. For the two distinct constraint settings, we propose novel deterministic algorithms grounded in an extended multilinear extension framework. Under matroid constraints, our algorithm achieves an approximation ratio of $(0.385 - ε)$, whereas for knapsack constraints, the proposed algorithm attains an approximation ratio of $(0.367 -ε)$. Both algorithms run in $\mathrm{poly}(n)$ query complexity, where $n$ is the size of the ground set, and improve upon the state-of-the-art deterministic approximation ratios of $(0.367 - ε)$ for matroid constraints and $0.25$ for knapsack constraints.

Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

Abstract

Submodular maximization constitutes a prominent research topic in combinatorial optimization and theoretical computer science, with extensive applications across diverse domains. While substantial advancements have been achieved in approximation algorithms for submodular maximization, the majority of algorithms yielding high approximation guarantees are randomized. In this work, we investigate deterministic approximation algorithms for maximizing non-monotone submodular functions subject to matroid and knapsack constraints. For the two distinct constraint settings, we propose novel deterministic algorithms grounded in an extended multilinear extension framework. Under matroid constraints, our algorithm achieves an approximation ratio of , whereas for knapsack constraints, the proposed algorithm attains an approximation ratio of . Both algorithms run in query complexity, where is the size of the ground set, and improve upon the state-of-the-art deterministic approximation ratios of for matroid constraints and for knapsack constraints.
Paper Structure (23 sections, 27 theorems, 102 equations, 1 table, 9 algorithms)

This paper contains 23 sections, 27 theorems, 102 equations, 1 table, 9 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Organization.
  4. Preliminary
  5. Polytope of Constraints.
  6. Extended Multilinear Extension.
  7. Lovász Extension.
  8. Matroid Constraints
  9. Deterministic-Pipage
  10. Dummy elements.
  11. Discrete Local Search
  12. Discrete Stationary Points.
  13. Deterministic Aided Continuous Greedy
  14. Split Algorithm.
  15. Continuous Greedy Algorithm.
  16. ...and 8 more sections

Key Result

Theorem 1

Given a non-negative submodular function $f:2^\mathcal{N}\rightarrow \mathbb{R}_{\ge 0}$ and a matroid $\mathcal{M} = (\mathcal{N}, \mathcal{I})$, Algorithm alg:matroid_main is a deterministic algorithm that uses $O_{\varepsilon}(n^5)$ queries, retu[34] A. Kulik, R. Schwartz, and H. Shachnai. A refi

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Definition 3: Random set
  • Definition 5: Coordinate-wise probabilistic sum
  • Definition 6: Marginal vector of $\mathbf{y}$
  • Lemma 10
  • proof
  • Theorem 11: Deterministic-Pipage, Theorem 5.6 in DBLP:conf/stoc/BuchbinderF25
  • Lemma 12
  • proof
  • ...and 42 more