Submodular Maximization with Matroid and Packing Constraints in Parallel

TL;DR

This paper introduces new parallel algorithms for submodular maximization under matroid and packing constraints, achieving near-optimal approximation ratios with significantly fewer adaptive rounds and parallel steps than previous methods.

Contribution

It presents the first low-adaptivity algorithms for submodular maximization with a matroid constraint and the first parallel algorithms for non-monotone submodular maximization under packing constraints.

Findings

Achieves a $1-1/e-\,\epsilon$ approximation for monotone functions with $O(\,\log^2 n/\epsilon^3)$ rounds.

Achieves a $1/e-\,\epsilon$ approximation for non-monotone functions with exponential speedup in adaptivity.

Provides parallel algorithms matching the state-of-the-art for packing LPs in terms of parallel rounds.

Abstract

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-\epsilon$ approximation for monotone functions and a $1/e-\epsilon$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/\epsilon^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-\epsilon$ approximation in $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.