Parallelized maximization of nonsubmodular function subject to a cardinality constraint

• Published in: Theoretical computer science Vol. 864; pp. 129 - 137
• Main Authors: Zhang, Hongxiang, Xu, Dachuan, Guo, Longkun, Tan, Jingjing
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 10.04.2021
• Subjects: Adaptive; Continuous generic submodularity ratio; Multilinear relaxation; Nonsubmodular; Continuous generic submodularity ratio; Adaptive; Multilinear relaxation; Nonsubmodular
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2021.02.035

•We devise parallel algorithms for non-submodular maximization based on rounding fractional solutions of its multilinear relaxation.•The developed techniques have the potential to inspire new algorithms with a provably low number of adaptive rounds.•The devised algorithm achieves performance guarantee close to the state-of-art result for the submodular version of the problem. In the paper, we consider the problem of maximizing the multilinear extension of a nonsubmodular set function subject to a k-cardinality constraint with adaptive rounds of evaluation queries. We devise an algorithm which achieves a ratio of (1−e−γ2−ϵ) and requires O(log⁡n/ϵ2) adaptive rounds and O(nlog⁡n/ϵ2) queries, where γ is the continuous generic submodularity ratio that compares favorably in flexibility to the traditional submodularity ratio proposed by Das and Kempe. The key idea of our algorithm is originated from the parallel-greedy algorithm proposed by Chekuri et al., but incorporating with two major changes to retain the performance guarantee: First, identify all good coordinates with the continuous generic submodularity ratio and gradient values approximately as large as the best coordinate, and increase along all these coordinates uniformly; Second, increase x along these coordinates by a dynamical increment whose value depends on γ. The key difficulty of our algorithm is that when the function is nonsubmodular, the set of the best coordinate does not decrease during iterations; while provided submodularity, the decreasing can be ensured by the parallel-greedy algorithm. Our algorithms slightly compromise performance guarantee for the sake of extending to constrained nonsubmodular maximization with parallelism, provided that the state-of-art algorithm for the corresponding submodular version attains an approximation ratio of (1−1/e−ϵ) and requires O(log⁡n/ϵ2) adaptive rounds.