An effective two‐level solution approach for the prize‐collecting generalized minimum spanning tree problem by iterated local search

Abstract

The prize-collecting generalized minimum spanning tree problem consists of finding a minimum cost spanning tree in an undirected graph, considering that the vertices are divided into clusters, that there is a nonnegative cost associated with each edge, that there is a prize to be collected on each vertex, and that only one vertex of each cluster belongs to the tree. Due to its NP-hardness, this problem remains a computational challenge even for small instances, and the practical applications that arise in telecommunication networks can attain large dimensions. The state-of-the-art algorithm finds good solutions for several instances; however, the quality of solutions and the computing time for large instances remain to be improved. We propose an iterated local search algorithm that works on a two-level solution approach, taking advantage of the structure of the problem. The first-level subproblem aims to select a vertex for each cluster, and the second-level subproblem addresses the determination of the minimum spanning tree covering such vertices. The performance of the resulting algorithm is evaluated with the most challenging instances from the literature, comparing the results with the state-of-the-art algorithm. The computational experiment conducted shows that, both in existing instances and in a new group of instances presented in this paper, the proposed algorithm outperforms the state-of-the-art algorithm.

Continuous approximation formulas for location problems

“…with probability one, which completes the proof since we can make 𝜖 arbitrarily small. The fact that our asymptotic bound (12) holds with probability one is due to the Borel-Cantelli lemma, because we have established that the probability of our event occurring is bounded above by a n for some a < 1. ▪ Proof of Theorem 22.…”

“…We draw points i.i.d. uniformly in the unit square and solve the generalized spanning tree problem using the prize collecting generalized spanning tree solver of Contreras-Bolton and Parada [12], letting each node simply carry the same prize. Other GMST solution methods can be found in [16,17,26,45].…”

Exact and heuristic solutions for the prize‐collecting geometric enclosure problem

“…vertex that is not included. For some recent works on prize-collecting problems, see Whittle et al (2022), Pantuza and de Souza (2022), Marzo andRibeiro (2020), andContreras-Bolton andParada (2021).…”

Dual-Neighborhood Search for Solving the Minimum Dominating Tree Problem

“…Carrabs et al [30] introduced a metaheuristic algorithm implementing a two-level structure to solve the shortest path problem for all colors. Contreras Bolton and Parada [31] proposed an iterative local search method to solve the generalized minimum spanning tree problem using a two-level solution.…”