Parallel Submodular Function Minimization

TL;DR

This paper introduces two new parallel algorithms for submodular function minimization, achieving different trade-offs between query complexity and parallel depth, and also presents a highly-parallel method for minimizing $ ext{l}_ ext{infinity}$-Lipschitz functions.

Contribution

The paper provides the first highly-parallel algorithms for SFM with novel depth-query trade-offs and introduces a new parallel approach for $ ext{l}_ ext{infinity}$-Lipschitz function minimization.

Findings

First method: depth 2, query complexity $n^{O(M)}$

Second method: depth $ ilde{O}(n^{1/3} M^{2/3})$, poly$(n, M)$ queries

New highly-parallel algorithm for $ ext{l}_ ext{infinity}$-Lipschitz function minimization

Abstract

We consider the parallel complexity of submodular function minimization (SFM). We provide a pair of methods which obtain two new query versus depth trade-offs a submodular function defined on subsets of $n$ elements that has integer values between $-M$ and $M$. The first method has depth $2$ and query complexity $n^{O(M)}$ and the second method has depth $\widetilde{O}(n^{1/3} M^{2/3})$ and query complexity $O(\mathrm{poly}(n, M))$. Despite a line of work on improved parallel lower bounds for SFM, prior to our work the only known algorithms for parallel SFM either followed from more general methods for sequential SFM or highly-parallel minimization of convex $\ell_2$-Lipschitz functions. Interestingly, to obtain our second result we provide the first highly-parallel algorithm for minimizing $\ell_\infty$-Lipschitz function over the hypercube which obtains near-optimal depth for obtaining constant accuracy.