Superpolynomial Smoothed Complexity of the 3-FLIP Algorithm for the Max-Cut Problem

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The 3-FLIP algorithm, a local search algorithm for the Max-Cut problem, can have a superpolynomial smoothed runtime, in contrast to previous results showing efficient smoothed runtimes for other local search algorithms.

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The paper presents two main results:

1. A new construction of graphs with O(n) vertices and maximum degree four, where the FLIP algorithm (a local search algorithm for the Max-Cut problem) takes exponential time to terminate, regardless of the pivot rule used. This provides a simpler construction compared to previous work.

2. The main result shows that the 3-FLIP algorithm, which moves up to 3 vertices across the cut in each step, can have a superpolynomial smoothed runtime. Specifically, the authors construct a graph with O(n) vertices where the smoothed runtime of the 3-FLIP algorithm can be as large as 2^(Ω(√n)). This is the first example of a local search algorithm for the Max-Cut problem whose smoothed runtime can be inefficient.

The key ideas are:

• Constructing a graph where the FLIP algorithm has an exponential runtime by triggering an exponential improving sequence.
• Leveraging linear combinations of uniform random variables to precisely control the local improvements during the execution of the 3-FLIP algorithm, allowing the construction of a graph with superpolynomial smoothed runtime.

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Belangrijkste Inzichten Gedestilleerd Uit

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

by Lukas Michel... om arxiv.org 04-05-2024

https://arxiv.org/pdf/2310.19594.pdf

Diepere vragen

How would the results change if the pivot rule for the 3-FLIP algorithm is restricted to not make moves where a single vertex would yield a higher improvement

If the pivot rule for the 3-FLIP algorithm is restricted to not make moves where a single vertex would yield a higher improvement, the results would likely change. This restriction would potentially lead to a more efficient algorithm in terms of smoothed runtime. By avoiding moves that do not provide the highest improvement, the algorithm could potentially converge faster to a local optimum. This adjustment could result in a more optimized and effective local search process, reducing the overall runtime and improving the algorithm's performance.

Are there other local search algorithms for the Max-Cut problem that can also exhibit superpolynomial smoothed runtimes, or is the 3-FLIP algorithm a unique case

While the 3-FLIP algorithm in the given context showcases superpolynomial smoothed runtimes, it is not necessarily a unique case. Other local search algorithms for the Max-Cut problem could also exhibit similar behaviors under certain conditions. The complexity of these algorithms in smoothed analysis can vary based on the specific characteristics of the problem instances and the algorithm's design. Therefore, it is possible that other local search algorithms for Max-Cut may also demonstrate superpolynomial smoothed runtimes in certain scenarios.

What are the implications of these results for the practical performance of local search algorithms for the Max-Cut problem, and how can they inform the design of more efficient algorithms

The results indicating superpolynomial smoothed runtimes for the 3-FLIP algorithm in the context of the Max-Cut problem have significant implications for the practical performance of local search algorithms. These findings suggest that under certain conditions, local search algorithms may not always perform efficiently, especially in the presence of random noise in the input data. This insight can inform the design and optimization of local search algorithms for Max-Cut, highlighting the importance of considering smoothed analysis to understand algorithm behavior more accurately. By incorporating these insights, researchers and practitioners can work towards developing more efficient algorithms that perform well in practice and provide reliable solutions for NP-hard problems like Max-Cut.