Submodular Function Maximization in Parallel via the Multilinear Relaxation

TL;DR

This paper develops parallel algorithms for maximizing the multilinear relaxation of monotone submodular functions under packing constraints, achieving near-optimal approximation ratios with poly-logarithmic adaptivity for various constraint types.

Contribution

It extends parallel submodular maximization techniques to general packing constraints using multilinear relaxation, providing algorithms with provable approximation guarantees.

Findings

Achieves a $(1-1/e- ext{epsilon})$-approximation in $O( ext{log}^2 m ext{log} n/ ext{epsilon}^4)$ rounds.

Provides randomized rounding schemes for fractional solutions applicable to multiple constraint types.

Demonstrates poly-logarithmic adaptivity for constraints like matroids, matchings, and knapsack.

Abstract

Balkanski and Singer [5] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Very recent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal $(1-1/e-\epsilon)$-approximation in $O(\log n/\epsilon^2)$ rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to packing constraints. Formally our problem is to maximize $F(x)$ over $x \in [0,1]^{n}$ subject to $Ax \le 1$ where $F$ is the multilinear relaxation of a monotone submodular function. Our algorithm achieves a near-optimal $(1-1/e-\epsilon)$-approximation in $O(\log^2 m \log n/\epsilon^4)$ rounds where $n$ is the cardinality of the ground set and $m$ is the number of packing constraints. For many constraints of interest, the resulting fractional solution can be rounded via known randomized rounding schemes that are oblivious to the specific submodular function. We thus derive randomized algorithms with poly-logarithmic adaptivity for a number of constraints including partition and laminar matroids, matchings, knapsack constraints, and their intersections.