This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian completion problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

中文翻译：

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无向图上哈密顿完备问题的多起点局部搜索算法

针对图论中的特定组合优化问题，本文提出了一种局部搜索算法：无向图上的哈密顿完备问题（HCP）。在此问题中，目标是为给定的无向图添加尽可能少的边以获得汉密尔顿图。主要在各种特定种类的无向图（例如树，单环图和串并图）的背景下研究了此问题。然而，所提出的算法专注于求解一般无向图的HCP。它可以被认为属于数学范畴，因为它将树木的精确线性时间解集成到了通用图的局部搜索算法中。这种集成利用了HCP和最小路径分区问题之间的紧密关系，这使得该算法对于解决后一个问题同样有用。此外，构造了一个问题实例的基准集，以证明所提出算法的质量。与最新解决方案的比较表明，该算法能够获得高质量的结果。