A near linear query lower bound for submodular maximization
Keywords: Submodular maximization, sublinear algorithm, query complexity, communication complexity
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] 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 (i.e.~$f(S) = \sum_{i \in S} w_i$ for unknown $w_i$s), 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 $\tilde{O}(n/k)$ time, and that this is tight up to polylog factors.
Primary Area: learning theory
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Submission Number: 5848
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