Abstract
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.
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References
Aneja YP (1980) An integer linear programming approach to the Steiner problem in graphs. Networks 10: 167–178
Balasundaram B, Butenko S (2006) Graph domination, coloring and cliques in telecommunications. In: Handbook of optimization in telecommunications. Springer, New York, pp 865–890
Butenko S, Cheng X, Du DZ, Pardalos PM (2002) On the construction of virtual backbone for ad-hoc wireless networks. In: Cooperative control: models, applications and algorithms. Kluwer, Dordrecht, pp 43–54
Chen S, Ljubić I, Raghavan S (2009) The regenerator location problem. Networks (to appear)
Chopra S, Gorres E, Rao MR (1992) Solving Steiner tree problem on a graph using branch and cut. ORSA J Comput 4(3): 320–335
Edmonds J (1971) Matroids and the greedy algorithm. Math Prog 1: 127–136
Fernandes ML, Gouveia L (1998) Minimal spanning trees with a constraint on the number of leaves. Eur J Oper Res 104: 250–261
Fujie T (2003) An exact algorithm for the maximum-leaf spanning tree problem. Comput Oper Res 30: 1931–1944
Fujie T (2004) The maximum-leaf spanning tree problem: formulations and facets. Networks 43(4): 212–223
Galbiati G, Maffioli F, Morzenti A (1994) A short note on the approximability of the maximum leaves spanning tree problem. Info Proc Lett 52: 45–49
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman, New York
Gouveia L, Simonetti L, Uchoa E (2009) Modelling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. Math Prog (published online)
Guha S, Khuller S (1998) Approximation algorithms for connected dominating sets. Algorithmica 20(4): 374–387
Koch T, Martin A (1998) Solving Steiner tree problems in graphs to optimality. Networks 33: 207–232
Lu H, Ravi R (1992) The power of local optimization: approximation algorithms for maximum-leaf spanning tree. In: Thirtieth annual allerton conference on communication, pp 533–542
Lu H, Ravi R (1998) Approximating maximum leaf spanning trees in almost linear time. J Algo 29: 132–141
Lucena A, Maculan N, Simonetti L (2008) Reformulations and solution algorithms for maximum leaf spanning tree problem. In: Abstracts of the VI ALIO/EURO workshop, pp 48–48
Magnanti TL, Wolsey LA (1995) Optimal trees. In: Network models. Handbooks in operations research and management science, vol 7. North Holland, Amsterdam, pp 503–615
Marathe MV, Breu H, Hunt III HB, Ravi SS, Rosenkrantz DJ (1995) Simple heuristics for unit disc graphs. Networks 25
Poggi de Arag⭠M, Uchoa E, Werneck R (2001) Dual heuristics on the exact solution of large Steiner problems. Electron Notes Discret Math 7:150–153
Polzin T, Daneshmand SV (2001) Improved algorithms for the Steiner problem in networks. Discret Appl Math 112(1-3): 263–300
Resende MGC, Pardalos PM (2006) Handbook of optimization in telecommunications. Springer, New York
Simonetti LG (2008) Otimização Combinatória: Problemas de Árvores Geradoras em Grafos. PhD thesis, PESC/COPPE, Universidade Federal do Rio de Janeiro (in Portuguese)
Solis-Oba S (1998) 2-approximation algorithm for finding a spanning tree with maximum number of leaves. Lect Notes Comput Sci 1461: 441–452
Wong R (1984) A dual ascent approach for Steiner tree problems on a directed graph. Math Prog 28: 271–287
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This paper is dedicated to Nicos Christofides, the former PhD supervisor of Abilio Lucena.
Abilio Lucena was partially funded by CNPq grant 473726/2007-6, Nelson Maculan was partially funded by CNPq grants 473420/2007-4 and 304301/2006-0, and Luidi Simonetti was partially funded by CNPq grant 140489/2004-5 and FAPESP grant 2008/01497-8.
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Lucena, A., Maculan, N. & Simonetti, L. Reformulations and solution algorithms for the maximum leaf spanning tree problem. Comput Manag Sci 7, 289–311 (2010). https://doi.org/10.1007/s10287-009-0116-5
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DOI: https://doi.org/10.1007/s10287-009-0116-5