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