Improved Lower Bounds for Submodular Function Minimization

TL;DR

This paper introduces a new technique to establish lower bounds for submodular function minimization, proving super-linear query complexity and near-optimal parallel round bounds, advancing understanding of the problem's computational limits.

Contribution

The paper presents a generic method for constructing submodular functions to derive lower bounds, establishing the first super-linear query lower bound and improving parallel complexity bounds for SFM.

Findings

Any deterministic SFM algorithm requires at least Ω(n log n) queries.

Parallel SFM algorithms need at least Ω(n / log n) rounds.

This work improves previous lower bounds and nearly settles the parallel complexity of SFM.

Abstract

We provide a generic technique for constructing families of submodular functions to obtain lower bounds for submodular function minimization (SFM). Applying this technique, we prove that any deterministic SFM algorithm on a ground set of $n$ elements requires at least $\Omega(n \log n)$ queries to an evaluation oracle. This is the first super-linear query complexity lower bound for SFM and improves upon the previous best lower bound of $2n$ given by [Graur et al., ITCS 2020]. Using our construction, we also prove that any (possibly randomized) parallel SFM algorithm, which can make up to $\mathsf{poly}(n)$ queries per round, requires at least $\Omega(n / \log n)$ rounds to minimize a submodular function. This improves upon the previous best lower bound of $\tilde{\Omega}(n^{1/3})$ rounds due to [Chakrabarty et al., FOCS 2021], and settles the parallel complexity of query-efficient SFM up to logarithmic factors due to a recent advance in [Jiang, SODA 2021].