Modelling and solving the perfect edge domination problem

A formulation is proposed for the perfect edge domination problem and some exact algorithms based on it are designed and tested. So far, perfect edge domination has been investigated mostly in computational complexity terms. Indeed, we could find no previous explicit mathematical formulation or exact algorithm for the problem. Furthermore, testing our algorithms also represented a challenge. Standard randomly generated graphs tend to contain a single perfect edge dominating solution, i.e., the trivial one, containing all edges in the graph. Accordingly, some quite elaborated procedures had to be devised to have access to more challenging instances. A total of 736 graphs were thus generated, all of them containing feasible solutions other than the trivial ones. Every graph giving rise to a weighted and a non weighted instance, all instances solved to proven optimality by two of the algorithms tested.Fil: do Forte, Vinicius L.. Universidade Federal Rural do Rio de Janeiro; BrasilFil: Lin, Min Chih. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Lucena, Abilio. Universidade Federal do Rio de Janeiro; BrasilFil: Maculan, Nelson. Universidade Federal do Rio de Janeiro; BrasilFil: Moyano, Verónica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Szwarcfiter, Jayme L.. Universidade do Estado de Rio do Janeiro; Brasil. Universidade Federal do Rio de Janeiro; Brasi

Exact solution approaches for the vehicle routing problem

Imperial Users onl

Reformulations and solution algorithms for the maximum leaf spanning tree problem

Given a graph G = (V, E), the maximum leaf spanning tree problem (MLSTP) is to find a spanning tree of G with as many leaves as possible. The problem is easy to solve when G is complete. However, for the general case, when the graph is sparse, it is proven to be NP-hard. In this paper, two reformulations are proposed for the problem. The first one is a reinforced directed graph version of a formulation found in the literature. The second recasts the problem as a Steiner arborescence problem over an associated directed graph. Branch-and-Cut algorithms are implemented for these two reformulations. Additionally, we also implemented an improved version of a MLSTP Branch-and-Bound algorithm, suggested in the literature. All of these algorithms benefit from pre-processing tests and a heuristic suggested in this paper. Computational comparisons between the three algorithms indicate that the one associated with the first reformulation is the overall best. It was shown to be faster than the other two algorithms and is capable of solving much larger MLSTP instances than previously attempted in the literature. © 2010 Springer-Verlag.73289311Aneja, Y.P., An integer linear programming approach to the Steiner problem in graphs (1980) Networks, 10, pp. 167-178Chopra, S., Gorres, E., Rao, M.R., Solving Steiner tree problem on a graph using branch and cut (1992) ORSA J Comput, 4 (3), pp. 320-335Edmonds, J., Matroids and the greedy algorithm (1971) Math Prog, 1, pp. 127-136Fernandes, M.L., Gouveia, L., Minimal spanning trees with a constraint on the number of leaves (1998) Eur J Oper Res, 104, pp. 250-261Fujie, T., An exact algorithm for the maximum-leaf spanning tree problem (2003) Comput Oper Res, 30, pp. 1931-1944Fujie, T., The maximum-leaf spanning tree problem: Formulations and facets (2004) Networks, 43 (4), pp. 212-223Galbiati, G., Maffioli, F., Morzenti, A., A short note on the approximability of the maximum leaves spanning tree problem (1994) Info Proc Lett, 52, pp. 45-49Garey, M.R., Johnson, D.S., (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness, , New York: W. H. FreemanGuha, S., Khuller, S., Approximation algorithms for connected dominating sets (1998) Algorithmica, 20 (4), pp. 374-387Koch, T., Martin, A., Solving Steiner tree problems in graphs to optimality (1998) Networks, 33, pp. 207-232Lu, H., Ravi, R., Approximating maximum leaf spanning trees in almost linear time (1998) J Algo, 29, pp. 132-141Poggi de Aragão, M., Uchoa, E., Werneck, R., Dual heuristics on the exact solution of large Steiner problems (2001) Electron Notes Discret Math, 7, pp. 150-153Polzin, T., Daneshmand, S.V., Improved algorithms for the Steiner problem in networks (2001) Discret Appl Math, 112 (1-3), pp. 263-300Resende, M.G.C., Pardalos, P.M., (2006) Handbook of Optimization in Telecommunications, , New York: SpringerSolis-Oba, S., 2-approximation algorithm for finding a spanning tree with maximum number of leaves (1998) Lect Notes Comput Sci, 1461, pp. 441-452Wong, R., A dual ascent approach for Steiner tree problems on a directed graph (1984) Math Prog, 28, pp. 271-28

Solving Diameter Constrained Minimum Spanning Tree Problems in Dense Graphs

International audienc

Uma heurística híbrida para o problema da árvore geradora de custo mínimo com restrição de diâmetro

International audienc

Lagrangian Based Heuristics for the Linear

this paper, two di#erent LOP heuristics are proposed and computationally tested. Heuristics are incorporated into a Lagrangian Relaxation (LR) framework and repeatedly called under Lagrangian modified arc costs. As a result, a series of feasible LOP solutions are generated which attempt to incorporating LR upper bounding information. Due to the exceedingly large number of inequalities that would have to be dualized to generate Lagrangian LOP upper bounds, a Relax and Cut algorithm, in the style proposed in [9, 10], is used. Computational experiments indicate that the approach appears highly e#ective for LOPs associated with the triangulation of input-output matrices. Indeed, for all 43 instances taken from LOLIB [13], optimal solutions have been generated. On the other hand, for only 2 out of the 30 instances proposed in [11], optimal solutions could not be generate

Heuristics for the diameter constrained minimum spanning tree problem

International audienc