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.
翻译:鉴于未指导的GG=(V,E) 美元,最长的路径问题(LIPP)包括获得最大基数子集 $W\subseeteq V$(LIPP), 以获得最大基数子子集 $W\subseq V$(W$) 美元, 以产生一个简单的G$路径。 在本文件中,我们提出了两个新的配方, 其制约指数指数数是问题, 以及解决问题的有效分支和切割程序。 第一个配方(cec) 是基于明确消除周期的制约, 第二个(c) 则确保通过切割限制确保连接。 我们从理论上和实验上将新提议的办法与最近提出的最先进的配方比较起来。 更具体地说, 我们用配方削减的多的配方和文献的配方数量是相同的。 此外,我们表明,这两种配方在理论上比塞克强。 我们用新的配方还提出了新的配方程序。 比较实验表明,新提议的配方c(c) 从理论角度来说,虽然不那么强,但从较强,我们从理论的观点看, 更强,最近提出的配法的配法的结果是最难的计算, 也就是在最初的计算,我们使用的计算, 进的计算方法可以用来在10级的计算, 进进进的计算到新式的计算到新式的计算到新式的计算, 进进进进进式的计算, 进到新式的计算到新式的计算到新式的计算。在10进进式的计算到新式的计算到新式的计算到新式的计算中, 进式的计算中,在新式的计算中, 进进进进的计算到新式的计算到新式的计算中, 进进进进进式的计算到新式的计算到新式的计算到新式的计算, 进式的计算到新式的计算到新式的计算中, 进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进的进进进进进进进的进的进的进的