In this paper, we provide the first deterministic algorithm that achieves 1/2-approximation for monotone submodular maximization subject to a knapsack constraint, while making a number of queries that scales only linearly with the size of the ground set n. Moreover, our result automatically paves the way for developing a linear-time deterministic algorithm that achieves the tight 1 − 1/e approximation guarantee for monotone submodular maximization under a cardinality (size) constraint. To complement our positive results, we also show strong information-theoretic lower bounds. More specifically, we show that when the maximum cardinality allowed for a solution is constant, no deterministic or randomized 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 Ω(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 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. Finally, we evaluate the performance of our algorithms on multiple real-life applications, including movie recommendation, location summarization, Twitter text summarization, and video summarization.
|Title of host publication||Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022|
|Editors||S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh|
|Publisher||Neural information processing systems foundation|
|State||Published - 2022|
|Event||36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States|
Duration: 28 Nov 2022 → 9 Dec 2022
|Name||Advances in Neural Information Processing Systems|
|Conference||36th Conference on Neural Information Processing Systems, NeurIPS 2022|
|Period||28/11/22 → 9/12/22|
Bibliographical noteFunding Information:
Funding in direct support of this work: The work of Moran Feldman was supported in part by Israel Science Foundation (ISF) grant no. 459/20. The work of Amin Karbasi was supported in part by the NSF (IIS-1845032), ONR (N00014-19-1-2406), and the AI Institute for Learning-Enabled Optimization at Scale (TILOS).
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