In this paper we study the Hamiltonian -median problem, in which we are given an edge-weighted graph and we are asked to determine vertex-disjoint cycles spanning all vertices of the graph and having minimum total weight. We introduce two new families of valid inequalities for a formulation of the problem in the space of edge variables. Each one of the families forbids solutions to the 2-factor relaxation of the problem that have less than cycles. The inequalities in one of the families are associated with large cycles of the underlying graph and generalize known inequalities associated with Hamiltonian cycles. The other family involves inequalities for the case with , associated with edge cuts and multi-cuts whose shores have specific cardinalities. We identify inequalities from both families that define facets of the polytope associated with the problem. We design branch-and-cut algorithms based on these families of inequalities and on inequalities associated with 2-opt moves removing sub-optimal solutions. Computational experiments on benchmark instances show that the proposed algorithms exhibit a comparable performance with respect to existing exact methods from the literature. Moreover the algorithms solve to optimality new instances with up to 400 vertices.

中文翻译：

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哈密​​顿 p 中值问题：多面体结果和分支割算法

在本文中，我们研究了哈密顿中值问题，其中给定了一个边加权图，并要求我们确定跨越该图所有顶点且具有最小总权重的顶点不相交循环。我们引入了两个新的有效不等式族来表述边缘变量空间中的问题。每个族都禁止解决小于周期的问题的二因子松弛问题。其中一族中的不等式与基础图的大循环相关，并概括了与哈密顿循环相关的已知不等式。另一个族涉及 的情况的不等式，与边缘切割和多重切割相关，其海岸具有特定的基数。我们从两个族中找出了不平等，这些不平等定义了与问题相关的多面体的各个方面。我们基于这些不等式族以及与消除次优解决方案的 2 步移动相关的不等式来设计分支剪切算法。对基准实例的计算实验表明，所提出的算法表现出与文献中现有的精确方法相当的性能。此外，该算法可求解最多 400 个顶点的新实例的最优性。