最长诱导路问题的新公式和分枝割算法
New formulations and branch-and-cut procedures for the longest induced path problem
摘要：
给定一个无向图$G=(V,E)$，最长诱导路径问题（LIPP）是求出一个最大基数子集$W\subseteq V$，使得$W$在$G$中诱导出一条简单路径.在本文中，我们提出了两个新的配方与指数数量的约束的问题，连同有效的分支和切割程序，其解决方案。第一个公式（cec）是基于明确消除循环的约束，第二个公式（cut）通过割集约束确保连通性。我们比较，在理论上和实验上，新提出的方法与最近在文献中提出的一个国家的最先进的配方。更具体地说，我们表明，多面体定义的配方切和在文献中的配方是相同的。此外，我们还证明了这两个公式在理论上比cec更强。我们还提出了一个新的分支和切割过程中使用的新配方。计算实验表明，新提出的配方cec，虽然从理论的角度来看，不太强，是最好的性能的方法，因为它可以解决所有，但在给定的时间限制内，在文献中使用的1065个基准实例之一。此外，我们新提出的方法在解决最优实例的中值时间方面优于最先进的公式。此外，我们进行了扩展的计算实验，考虑到更具挑战性和难以解决的较大的情况下，并评估结果的影响时，提供初始可行的解决方案（热启动）的配方。
英文摘要：
Given an undirected graph $G=(V,E)$, the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset $W\subseteq V$ such that $W$ induces a simple path in $G$. In this paper, we propose two new formulations with an exponential number of constraints for the problem, together with effective branch-and-cut procedures for its solution. While the first formulation (cec) is based on constraints that explicitly eliminate cycles, the second one (cut) ensures connectivity via cutset constraints. We compare, both theoretically and experimentally, the newly proposed approaches with a state-of-the-art formulation recently proposed in the literature. More specifically, we show that the polyhedra defined by formulation cut and that of the formulation available in the literature are the same. Besides, we show that these two formulations are stronger in theory than cec. We also propose a new branch-and-cut procedure using the new formulations. Computational experiments show that the newly proposed formulation cec, although less strong from a theoretical point of view, is the best performing approach as it can solve all but one of the 1065 benchmark instances used in the literature within the given time limit. In addition, our newly proposed approaches outperform the state-of-the-art formulation when it comes to the median times to solve the instances to optimality. Furthermore, we perform extended computational experiments considering more challenging and hard-to-solve larger instances and evaluate the impacts on the results when offering initial feasible solutions (warm starts) to the formulations.
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