Some easy optimization problems have the overlap-gap property

Shuangping Li, Tselil Schramm
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:3582-3622, 2025.

Abstract

We show that the shortest $s$-$t$ path problem has the overlap-gap property in (i) sparse $\mathbb{G}(n,p)$ graphs and (ii) complete graphs with i.i.d. Exponential edge weights. Furthermore, we demonstrate that in sparse $\mathbb{G}(n,p)$ graphs, shortest path is solved by $O(\log n)$-degree polynomial estimators, and a uniform approximate shortest path can be sampled in polynomial time. This constitutes the first example in which the overlap-gap property is not predictive of algorithmic intractability for a (non-algebraic) average-case optimization problem.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-li25b, title = {Some easy optimization problems have the overlap-gap property}, author = {Li, Shuangping and Schramm, Tselil}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {3582--3622}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/li25b/li25b.pdf}, url = {https://proceedings.mlr.press/v291/li25b.html}, abstract = {We show that the shortest $s$-$t$ path problem has the overlap-gap property in (i) sparse $\mathbb{G}(n,p)$ graphs and (ii) complete graphs with i.i.d. Exponential edge weights. Furthermore, we demonstrate that in sparse $\mathbb{G}(n,p)$ graphs, shortest path is solved by $O(\log n)$-degree polynomial estimators, and a uniform approximate shortest path can be sampled in polynomial time. This constitutes the first example in which the overlap-gap property is not predictive of algorithmic intractability for a (non-algebraic) average-case optimization problem.} }
Endnote
%0 Conference Paper %T Some easy optimization problems have the overlap-gap property %A Shuangping Li %A Tselil Schramm %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-li25b %I PMLR %P 3582--3622 %U https://proceedings.mlr.press/v291/li25b.html %V 291 %X We show that the shortest $s$-$t$ path problem has the overlap-gap property in (i) sparse $\mathbb{G}(n,p)$ graphs and (ii) complete graphs with i.i.d. Exponential edge weights. Furthermore, we demonstrate that in sparse $\mathbb{G}(n,p)$ graphs, shortest path is solved by $O(\log n)$-degree polynomial estimators, and a uniform approximate shortest path can be sampled in polynomial time. This constitutes the first example in which the overlap-gap property is not predictive of algorithmic intractability for a (non-algebraic) average-case optimization problem.
APA
Li, S. & Schramm, T.. (2025). Some easy optimization problems have the overlap-gap property. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:3582-3622 Available from https://proceedings.mlr.press/v291/li25b.html.

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