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arXiv · 1903.12112

Merging Combinatorial Design and Optimization: the Oberwolfach Problem

Fabio Salassa·Gabriele Dragotto·Tommaso Traetta·Marco Buratti·Federico Della Croce
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Abstract

The Oberwolfach Problem $OP(F)$ -- posed by Gerhard Ringel in 1967 -- is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In this paper, we provide all the necessary equipment to generate solutions to $OP(F)$ for relatively small orders by using the so-called difference methods. From the theoretical standpoint, we present new insights on the combinatorial structures involved in the solution of the problem. Computationally, we provide a full recipe whose base ingredients are advanced optimization models and tailored algorithms. This algorithmic arsenal can solve the $OP(F)$ for all possible orders up to $60$ with the modest computing resources of a personal computer. The new $20$ orders, from $41$ to $60$, encompass $241200$ instances of the Oberwolfach Problem, which is 22 times greater than those solved in previous contributions.

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Fabio Salassa, Gabriele Dragotto, Tommaso Traetta, Marco Buratti, Federico Della Croce. 2020-11-10. Merging Combinatorial Design and Optimization: the Oberwolfach Problem. https://arxiv.org/abs/1903.12112

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