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

TL;DR

This paper introduces a new parallel algorithm for non-monotone submodular maximization under knapsack constraints, improving approximation factors while maintaining low adaptive complexity, validated through extensive experiments.

Contribution

It presents a novel alternate threshold framework that enhances solution quality without increasing adaptive complexity, improving the approximation factor from 8+ε to 7+ε.

Findings

Improved approximation factor from 8+ε to 7+ε.

Achieved low adaptive complexity of O(log n).

Demonstrated superior solution quality in practical applications.

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 $8+\epsilon$ to $7+\epsilon$ 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.