We study the problem of maximizing a nonmonotone submodular function subject to a cardinality constraint in the streaming model. Our main contributions are two singlepass (semi)streaming algorithms that use $\tilde{O}(k)\cdot\mathrm{poly}(1/\varepsilon)$ memory, where $k$ is the size constraint. At the end of the stream, both our algorithms postprocess their data structures using any offline algorithm for submodular maximization, and obtain a solution whose approximation guarantee is $\frac{\alpha}{1+\alpha}\varepsilon$, where $\alpha$ is the approximation of the offline algorithm. If we use an exact (exponential time) postprocessing algorithm, this leads to $\frac{1}{2}\varepsilon$ approximation (which is nearly optimal). If we postprocess with the algorithm of BuchbinderFeldman '19, that achieves the stateoftheart offline approximation guarantee of $\alpha=0.385$, we obtain $0.2779$approximation in polynomial time, improving over the previously best polynomialtime approximation of $0.1715$ due to Feldman'18. One of our algorithms is combinatorial and enjoys fast update and overall running times. Our other algorithm is based on the multilinear extension, enjoys an improved space complexity, and can be made deterministic in some settings of interest.
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Streaming Algorithm for Monotone kSubmodular Maximization with Cardinality Constraints
Maximizing a monotone ksubmodular function subject to cardinality constraints is a general model for several applications ranging from influence maximization with multiple products to sensor placement with multiple sensor types and online ad allocation. Due to the large problem scale in many applications and the online nature of ad allocation, a need arises for algorithms that process elements in a streaming fashion and possibly make online decisions. In this work, we develop a new streaming algorithm for maximizing a monotone ksubmodular function subject to a percoordinate cardinality constraint attaining an approximation guarantee close to the state of the art guarantee in the offline setting. Though not typical for streaming algorithms, our streaming algorithm also readily applies to the online setting with free disposal. Our algorithm is combinatorial and enjoys fast running time and small number of function evaluations. Furthermore, its guarantee improves as the cardinality constraints get larger, which is especially suited for the large scale applications. For the special case of maximizing a submodular function with large budgets, our combinatorial algorithm matches the guarantee of the stateoftheart continuous algorithm, which requires significantly more time and function evaluations.
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 Award ID(s):
 1750716
 NSFPAR ID:
 10389490
 Date Published:
 Journal Name:
 Proceedings of Machine Learning Research
 Volume:
 162
 ISSN:
 26403498
 Page Range / eLocation ID:
 59445967
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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