Submodular Maximization in Clean Linear Time

• URL: http://arxiv.org/abs/2006.09327v5
• Date: Tue, 12 Apr 2022 11:23:51 GMT
• Title: Submodular Maximization in Clean Linear Time
• Authors: Wenxin Li, Moran Feldman, Ehsan Kazemi, Amin Karbasi
• Abstract summary: We provide the first deterministic algorithm that achieves the tight $1-1/e$ approximation guarantee for submodularimation under a cardinality (size) constraint. We show that when the cardinality allowed for a solution is constant, no algorithm making a sub-linear number of function evaluations can guarantee any constant ratio. We extensively evaluate our algorithms on multiple real-life machine learning applications, including movie recommendation, location summarization, twitter text summarization and video summarization.
• Score: 42.51873363778082
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we provide the first deterministic algorithm that achieves the tight $1-1/e$ approximation guarantee for submodular maximization under a cardinality (size) constraint while making a number of queries that scales only linearly with the size of the ground set $n$. To complement our result, we also show strong information-theoretic lower bounds. More specifically, we show that when the maximum cardinality allowed for a solution is constant, no algorithm making a sub-linear number of function evaluations can guarantee any constant approximation ratio. Furthermore, when the constraint allows the selection of a constant fraction of the ground set, we show that any algorithm making fewer than $\Omega(n/\log(n))$ function evaluations cannot perform better than an algorithm that simply outputs a uniformly random subset of the ground set of the right size. We then provide a variant of our deterministic algorithm for the more general knapsack constraint, which is the first linear-time algorithm that achieves $1/2$-approximation guarantee for this constraint. Finally, we extend our results to the general case of maximizing a monotone submodular function subject to the intersection of a $p$-set system and multiple knapsack constraints. We extensively evaluate the performance of our algorithms on multiple real-life machine learning applications, including movie recommendation, location summarization, twitter text summarization and video summarization.

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