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Improved Parallel Algorithm for Non-Monotone Submodular Maximization under Knapsack Constraint

Tan D. Tran, Canh V. Pham, Dung T. K. Ha, Phuong N. H. Pham

TL;DR

The paper tackles non-monotone submodular maximization under a knapsack constraint (SMK) on large ground sets. It introduces the AST algorithm, a parallel approach that attains a $(7+\epsilon)$-approximation within $O(\log n)$ adaptive rounds and $ ilde{O}(nk)$ queries by employing an alternate threshold framework that alternates two disjoint candidate solutions across a constant number of iterations. The method boosts solution quality via a threshold-sampling-based process and a boosting phase that may incorporate UnSubMax, leading to strong theoretical guarantees and practical performance. Empirical evaluations on Revenue Maximization, Image Summarization, and Maximum Weighted Cut demonstrate superior objective values and comparable adaptivity relative to state-of-the-art parallel SMK algorithms, highlighting AST's potential for scalable, near-optimal optimization in large-scale settings.

Abstract

This work proposes an efficient parallel algorithm for non-monotone submodular maximization under a knapsack constraint problem over the ground set of size $n$. Our algorithm improves the best approximation factor of the existing parallel one from $8+ε$ to $7+ε$ with $O(\log n)$ adaptive complexity. The key idea of our approach is to create a new alternate threshold algorithmic framework. This strategy alternately constructs two disjoint candidate solutions within a constant number of sequence rounds. Then, the algorithm boosts solution quality without sacrificing the adaptive complexity. Extensive experimental studies on three applications, Revenue Maximization, Image Summarization, and Maximum Weighted Cut, show that our algorithm not only significantly increases solution quality but also requires comparative adaptivity to state-of-the-art algorithms.

Improved Parallel Algorithm for Non-Monotone Submodular Maximization under Knapsack Constraint

TL;DR

The paper tackles non-monotone submodular maximization under a knapsack constraint (SMK) on large ground sets. It introduces the AST algorithm, a parallel approach that attains a -approximation within adaptive rounds and queries by employing an alternate threshold framework that alternates two disjoint candidate solutions across a constant number of iterations. The method boosts solution quality via a threshold-sampling-based process and a boosting phase that may incorporate UnSubMax, leading to strong theoretical guarantees and practical performance. Empirical evaluations on Revenue Maximization, Image Summarization, and Maximum Weighted Cut demonstrate superior objective values and comparable adaptivity relative to state-of-the-art parallel SMK algorithms, highlighting AST's potential for scalable, near-optimal optimization in large-scale settings.

Abstract

This work proposes an efficient parallel algorithm for non-monotone submodular maximization under a knapsack constraint problem over the ground set of size . Our algorithm improves the best approximation factor of the existing parallel one from to with adaptive complexity. The key idea of our approach is to create a new alternate threshold algorithmic framework. This strategy alternately constructs two disjoint candidate solutions within a constant number of sequence rounds. Then, the algorithm boosts solution quality without sacrificing the adaptive complexity. Extensive experimental studies on three applications, Revenue Maximization, Image Summarization, and Maximum Weighted Cut, show that our algorithm not only significantly increases solution quality but also requires comparative adaptivity to state-of-the-art algorithms.
Paper Structure (19 sections, 8 theorems, 35 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 19 sections, 8 theorems, 35 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. New technical approach.
  4. Related Works
  5. Preliminaries
  6. Unconstrained Submodular Maximization ($\mathsf{UnSubMax}$)
  7. Density Threshold (DS).
  8. Proposed Algorithm
  9. Experimental Evaluation
  10. Revenue Maximization ($\mathsf{RM}$).
  11. Maximum Weighted Cut ($\mathsf{MWC}$).
  12. Image Summarization ($\mathsf{IS}$).
  13. Experimental Result.
  14. Conclusions
  15. $\mathsf{RandBatch}$ Algorithm
  16. ...and 4 more sections

Key Result

Lemma 1

The sets $A$, $L$ output by $\mathsf{RandBatch}(\theta, I, M, \epsilon, f(\cdot), c(\cdot))$ satisfy ${\mathbb E}[f(A)]\geq (1-\epsilon)^2\theta \cdot {\mathbb E}[c(A)]$ and $\epsilon \cdot M \cdot \sum_{u\in L}f(u|A)\leq {\mathsf{OPT}}$.

Figures (1)

  • Figure 1: Performance of algorithms for $\mathsf{\hbox{non-monotone} SMK}$ on three instances: (a), (b) Revenue Maximization; (c), (d) Image Summarization and (e), (f) Maximum Weighted Cut. The budget values represent fractions of the total cost of all elements.

Theorems & Definitions (11)

  • Definition 1: Adaptive complexity or Adaptivity BalkanskiS18-adapt-stoc18
  • Lemma 1: Lemma 1 in Cui-aaai23
  • Lemma 2: Lemma 2 in Cui-aaai23
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • Theorem 4.1
  • proof
  • ...and 1 more