A tabu search algorithm for the minmax regret minimum spanning tree problem with interval data

Abstract

This paper deals with the strongly NP-hard minmax regret version of the minimum spanning tree problem with interval costs. The best known exact algorithms solve the problem in reasonable time for rather small graphs. In this paper an algorithm based on the idea of tabu search is constructed. Some properties of the local minima are shown. Exhaustive computational tests for various classes of graphs are performed. The obtained results suggest that the proposed tabu search algorithm quickly outputs optimal solutions for the smaller instances, previously discussed in the existing literature. Furthermore, some arguments that this algorithm performs well also for larger instances are provided.

Networks, uncertainty, applications and a crusade for optimality

“…With respect to the heuristic approaches for the MMR-ST, three classes of algorithms are found in the literature: (i) the two "one-scenario" heuristics HM and HU, the first proposed in [Kasperski and Zieliński, 2006] (where it is shown that it has an approximation ratio 2) and the second proposed in [Montemmani et al, 2007]; (ii) a simulated annealing (SA) proposed in [Nikulin, 2008]; and (iii) a tabu search (KMZ-TS) proposed in [Kasperski et al, 2012]. The SA approach was applied to small instances of the MMR-ST (up to 30 nodes) and reasonable results were obtained; the author pointed out that the approach should also work properly for large instances due to the search scheme used in the algorithm.…”

“…The SA approach was applied to small instances of the MMR-ST (up to 30 nodes) and reasonable results were obtained; the author pointed out that the approach should also work properly for large instances due to the search scheme used in the algorithm. In [Kasperski et al, 2012], the KMZ-TS algorithm was extensively tested on different sets of instances (Ya, He1, Mo and others), and it is shown that its performance is remarkably better than the one reported for the SA.…”

“…More precisely, a BD and a branch-and-cut approach are designed to solve benchmark instances that extend the size of instances for which an exact algorithm gets optimal solutions or small gaps. Additionally, the obtained lower bounds allow to improve the knowledge about the quality of the solutions given by the approach proposed in [Kasperski et al, 2012].…”

“…The solution obtained for the scenario s + defined by the upper limits of the intervals, i.e., c s + e = c + e , has also shown an interesting performance [psee, e.g., Montemmani et al, 2007, Kasperski, 2008, Kasperski et al, 2012, although it has been proved that this solution can be arbitrarily bad [see Kasperski, 2008]. Both solutions, X M and X + , will be used as part of the exact approaches proposed in this work.…”

“…Only for few problems, the corresponding MMR counterpart is polynomially solvable [see Candia-Véjar et al, 2011]. Several exact and heuristic approaches have been proposed for different MMR problems including MMR-ST [Yaman et al, 2001, Montemmani and Gambardella, 2005, Montemmani, 2006, Nikulin, 2008, Kasperski, 2008, Kasperski et al, 2012, MMR-P [Karasan et al, 2004, Kasperski, 2008, MMR-A [Pereira and Averbakh, 2011a], MMR Set Covering [Pereira and Averbakh, 2011b], MMR-TSP [Montemmani et al, 2007].…”