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Fast Approximation Algorithm for Non-Monotone DR-submodular Maximization under Size Constraint

Tan D. Tran, Canh V. Pham

TL;DR

This work studies non-monotone DR-submodular maximization on the integer lattice under a size constraint and presents two deterministic near-linear-time algorithms, FastDrSub and FastDrSub+. FastDrSub achieves a constant $0.044$-approximation with $O(n \log k)$ oracle queries, while FastDrSub+ reaches a near $\tfrac{1}{4}$-approximation with $O\left(\frac{n}{\epsilon}\log\left(\frac{1}{\epsilon}\right)\log k\right)$ queries for any $\epsilon>0$, both built on a reduction to standard submodular knapsack optimization. The methods are shown to be efficient in practice, closely competitive with or surpassing state-of-the-art baselines in Revenue Maximization tasks while drastically reducing the number of oracle queries. The algorithms combine a novel two-vector construction with a thresholding/greedy framework, and the experimental results corroborate their practical impact for large-scale DR-submodular problems. Overall, the paper advances deterministic near-linear algorithms for DR-submodular maximization under size constraints and broadens the toolkit for scalable submodular optimization on the integer lattice.

Abstract

This work studies the non-monotone DR-submodular Maximization over a ground set of $n$ subject to a size constraint $k$. We propose two approximation algorithms for solving this problem named FastDrSub and FastDrSub++. FastDrSub offers an approximation ratio of $0.044$ with query complexity of $O(n \log(k))$. The second one, FastDrSub++, improves upon it with a ratio of $1/4-ε$ within query complexity of $(n \log k)$ for an input parameter $ε>0$. Therefore, our proposed algorithms are the first constant-ratio approximation algorithms for the problem with the low complexity of $O(n \log(k))$. Additionally, both algorithms are experimentally evaluated and compared against existing state-of-the-art methods, demonstrating their effectiveness in solving the Revenue Maximization problem with DR-submodular objective function. The experimental results show that our proposed algorithms significantly outperform existing approaches in terms of both query complexity and solution quality.

Fast Approximation Algorithm for Non-Monotone DR-submodular Maximization under Size Constraint

TL;DR

This work studies non-monotone DR-submodular maximization on the integer lattice under a size constraint and presents two deterministic near-linear-time algorithms, FastDrSub and FastDrSub+. FastDrSub achieves a constant -approximation with oracle queries, while FastDrSub+ reaches a near -approximation with queries for any , both built on a reduction to standard submodular knapsack optimization. The methods are shown to be efficient in practice, closely competitive with or surpassing state-of-the-art baselines in Revenue Maximization tasks while drastically reducing the number of oracle queries. The algorithms combine a novel two-vector construction with a thresholding/greedy framework, and the experimental results corroborate their practical impact for large-scale DR-submodular problems. Overall, the paper advances deterministic near-linear algorithms for DR-submodular maximization under size constraints and broadens the toolkit for scalable submodular optimization on the integer lattice.

Abstract

This work studies the non-monotone DR-submodular Maximization over a ground set of subject to a size constraint . We propose two approximation algorithms for solving this problem named FastDrSub and FastDrSub++. FastDrSub offers an approximation ratio of with query complexity of . The second one, FastDrSub++, improves upon it with a ratio of within query complexity of for an input parameter . Therefore, our proposed algorithms are the first constant-ratio approximation algorithms for the problem with the low complexity of . Additionally, both algorithms are experimentally evaluated and compared against existing state-of-the-art methods, demonstrating their effectiveness in solving the Revenue Maximization problem with DR-submodular objective function. The experimental results show that our proposed algorithms significantly outperform existing approaches in terms of both query complexity and solution quality.

Paper Structure

This paper contains 17 sections, 5 theorems, 37 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions and Techniques.
  3. Organization
  4. Related Works
  5. Monotone DR-submodular functions.
  6. Non-monotone DR-submodular functions.
  7. Preliminaries
  8. The Proposed Algorithms
  9. Fast Approximation Algorithm (FastDrSub)
  10. Fast Approximation Plus Algorithm (FastDrSub+)
  11. Experimental Evaluation
  12. Experiment Settings
  13. Compared Algorithms
  14. Applications and Datasets
  15. Other settings.
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

For any $\textbf{s} \in \mathbb{Z}^E_+$ and two vectors $\textbf{x}, \textbf{y} \in \mathbb{Z}^E_+$ such that $\textbf{x} \wedge \textbf{y}=\textbf{0}$ we have

Figures (3)

  • Figure 1: Performance of algorithms on Revenue Maximization for the Facebook dataset: (a) The objective values, (b) The number of queries and (c) their running time.
  • Figure 2: Performance of algorithms on Revenue Maximization for the AstroPh dataset: (a) The objective values, (b) The number of queries and (c) their running time.
  • Figure 3: Performance of algorithms on Revenue Maximization for the Enron dataset: (a) The objective values, (b) The number of queries and (c) their running time.

Theorems & Definitions (11)

  • definition thmcounterdefinition: $\mathsf{DrSMC}$ problem
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • theorem 2
  • ...and 1 more