Parallelized Maximization of Nonsubmodular Function Subject to a Cardinality Constraint

Abstract

In the paper, we study the adaptivity of maximizing a monotone nonsubmodular function subject to a cardinality constraint. Adaptive approximation algorithm has been previously developed for the similar constrained maximization problem against submodular function, attaining an approximation ratio of $\left( 1-1/e-\epsilon \right)$ and $O\left( \log n/\epsilon ^{2}\right)$ rounds of adaptivity. For more general constraints, Chandra and Kent described parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to either cardinality or packing constraints, achieving a near-optimal $\left( 1-1/e-\epsilon \right)$-approximation in $O\left( \log ^{2}m\log n/\epsilon ^{4}\right)$ rounds. We propose an Expand-Parallel-Greedy algorithm for the multilinear relaxation of a monotone and normalized set function subject to a cardinality constraint based on rounding the multilinear relaxation of the function. The algorithm achieves a ratio of $\left( 1-e^{-\gamma ^{2}}-\epsilon \right)$, runs in $O\left( \log n/\epsilon ^{2}\right)$ adaptive rounds and requires $O\left( \left( n\log n/\epsilon ^{2}\right) \right)$ queries, where $\gamma$ is the Continuous generic submodularity ratio.