Energy landscapes of some matching-problem ensembles

Till Kahlke; Alexander K. Hartmann

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arXiv · 2304.01100

Energy landscapes of some matching-problem ensembles

Till Kahlke·Alexander K. Hartmann

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Abstract

The maximum-weight matching problem and the behavior of its energy landscape is numerically investigated. We apply a perturbation method adapted from the analysis of spin glasses. This gives inside into the complexity of the energy landscape of different ensembles. Erd\"os-Renyi graphs and ring graphs with randomly added edges are considered and two types of distributions for the random edge weighs are used. For maximum-weight matching, fast and scalable algorithms exist, such that we can study large graphs of more than $10^5$ nodes. Our results show that the structure of the energy landscape for standard ensembles of matching is simple, comparable to the energy landscape of a ferromagnet. Nonetheless, for some of the here presented ensembles our results allow for the presence of complex energy landscapes in the spirit of Replica-Symmetry Breaking.

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Till Kahlke, Alexander K. Hartmann. 2023-04-03. Energy landscapes of some matching-problem ensembles. https://arxiv.org/abs/2304.01100

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