Boosting Parallel Algorithms in Linear Queries for Non-Monotone Submodular Maximization

Canh V. Pham, Tan D. Tran, Dung T.K. Ha, My T. Thai

Keywords: Submodular Maximization, Non-Monotone, Parallel Algorithms

TL;DR: We propose two efficient parallel algorithms that improve both the approximation ratio and query complexity of existing practical parallel algorithms for the non-monotone submodular maximization

Abstract:

In this work, we propose two efficient parallel algorithms, $\mathsf{LinAst}$ and $\mathsf{LinAtg}$, that improve both the approximation ratio and query complexity of existing practical parallel algorithms for the non-monotone submodular maximization over the ground set of sized $n$ under a cardinality constraint $k$. Specifically, our algorithms keep the best adaptive complexity of $O(\log n)$ while significantly improving the approximation ratio from $1/6-\epsilon$ to $0.193-\epsilon$ and reducing the query complexity from $O(n\log (k))$ to $O(n)$.
The key building block of our algorithms is $\mathsf{LinAdapt}$, a constant approximation ratio within $O(\log n)$ sequence rounds and linear queries. $\mathsf{LinAdapt}$ can reduce the query complexity by providing $O(1)$ guesses of the optimal value. We further introduce the $\mathsf{BoostAdapt}$ algorithm returning a better ratio of $1/4-\epsilon$ within $O(\log (n)\log (k))$ adaptive complexity and $O(n\log (k))$ query complexity. Our $\mathsf{BoostAdapt}$ works in a novel staggered greedy threshold framework that alternately constructs two disjoint sets in $O(\log k)$ sequential rounds. Besides theoretical analysis, the experiment results on validated benchmarks confirm the superiority of our algorithms in terms of solution quality, the number of required queries, and running time over cutting-edge algorithms.

Primary Area: optimization

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Submission Number: 8680