A Near Linear Query Lower Bound for Submodular Maximization

Binghui Peng, Aviad Rubinstein
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:48852-48871, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-peng25d, title = {A Near Linear Query Lower Bound for Submodular Maximization}, author = {Peng, Binghui and Rubinstein, Aviad}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {48852--48871}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/peng25d/peng25d.pdf}, url = {https://proceedings.mlr.press/v267/peng25d.html}, 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.} }
Endnote
%0 Conference Paper %T A Near Linear Query Lower Bound for Submodular Maximization %A Binghui Peng %A Aviad Rubinstein %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-peng25d %I PMLR %P 48852--48871 %U https://proceedings.mlr.press/v267/peng25d.html %V 267 %X 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.
APA
Peng, B. & Rubinstein, A.. (2025). A Near Linear Query Lower Bound for Submodular Maximization. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:48852-48871 Available from https://proceedings.mlr.press/v267/peng25d.html.

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