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arXiv · 2310.19594

Superpolynomial smoothed complexity of 3-FLIP in Local Max-Cut

Lukas Michel·Alex Scott
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Abstract

Local search algorithms for NP-hard problems such as Max-Cut frequently perform much better in practice than worst-case analysis suggests. Smoothed analysis has proved an effective approach to understanding this: a substantial literature shows that when a small amount of random noise is added to input data, local search algorithms typically run in polynomial or quasi-polynomial time. In this paper, we provide the first example where a local search algorithm for the Max-Cut problem fails to be efficient in the framework of smoothed analysis. Specifically, we construct a graph with $n$ vertices where the smoothed runtime of the 3-FLIP algorithm can be as large as $2^{\Omega(\sqrt{n})}$. Additionally, for the setting without random noise, we give a new construction of graphs where the runtime of the FLIP algorithm is $2^{\Omega(n)}$ for any pivot rule. These graphs are much smaller and have a simpler structure than previous constructions.

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Lukas Michel, Alex Scott. 2023-10-30. Superpolynomial smoothed complexity of 3-FLIP in Local Max-Cut. https://arxiv.org/abs/2310.19594

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