A near Linear Query Lower Bound for Submodular Maximization

Abstract

We revisit the problem of selecting $k$-out-of-$n$ elements with the goal of optimizing an objective function, and ask whether it can be solved approximately with sublinear query complexity. For objective functions that are monotone submodular, [Li, Feldman, Kazemi, Karbasi, NeurIPS’22; Kuhnle, AISTATS’21] gave an $\Omega(n/k)$ query lower bound for approximating to within any constant factor. We strengthen their lower bound to a nearly tight $\tilde{\Omega}(n)$. This lower bound holds even for estimating the value of the optimal subset. When the objective function is additive, we prove that finding an approximately optimal subset still requires near-linear query complexity, but we can estimate the value of the optimal subset in $\tilde{O}(n/k)$ queries, and that this is tight up to polylog factors.