An Optimal Streaming Algorithm for Submodular Maximization with a Cardinality Constraint

• Published in: Mathematics of operations research Vol. 47; no. 4; pp. 2667 - 2690
• Main Authors: Alaluf, Naor, Ene, Alina, Feldman, Moran, Nguyen, Huy L., Suh, Andrew
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.11.2022; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; Approximation; cardinality constraint; Combinatorial analysis; Constraint modelling; Data structures; Mathematical analysis; Mathematical functions; Maximization; Operations research; Optimization; Polynomials; Primary: 90C27; secondary: 68W27; semi-streaming algorithms; submodular maximization
• ISSN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2021.1224

We study the problem of maximizing a nonmonotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi) streaming algorithm that uses roughly  O ( k / ε 2 )  memory, where k is the size constraint. At the end of the stream, our algorithm postprocesses its data structure using any off-line algorithm for submodular maximization and obtains a solution whose approximation guarantee is  α / ( 1 + α ) − ε , where α is the approximation of the off-line algorithm. If we use an exact (exponential time) postprocessing algorithm, this leads to  1 / 2 − ε  approximation (which is nearly optimal). If we postprocess with the state-of-the-art offline approximation algorithm, whose guarantee is  α = 0.385 , we obtain a 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715. It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for nonmonotone submodular maximization, and enjoys a fast update time of  O ( ε −2 ( log k + log ( 1 + α ) ) )  per element.