Abstract
Given a treeG = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the “rooted subtree problem”, is to find a maximum weight subtree, with a specified root, from a given set of subtrees.
The second problem, called “the subtree packing problem”, is to find a maximum weight packing of node disjoint subtrees chosen from a given set of subtrees, where the value of each subtree may depend on its root.
We show that the complexity status of both problems is related, and that the subtree packing problem is polynomial if and only if each rooted subtree problem is polynomial. In addition we show that the convex hulls of the feasible solutions to both problems are related: the convex hull of solutions to the packing problem is given by “pasting together” the convex hulls of the rooted subtree problems.
We examine in detail the case where the set of feasible subtrees rooted at nodei consists of all subtrees with at mostk nodes. For this case we derive valid inequalities, and specify the convex hull whenk ⩽ 4.
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References
E.H. Aghezzaf, “Optimal constrained rooted subtrees and partitioning problems on tree graphs,” Doctoral thesis, Faculté des Sciences, Université Catholique de Louvain (Louvain-la-Neuve, Belgium, 1992).
E.H. Aghezzaf, T.L. Magnanti and L.A. Wolsey, “Optimizing constrained subtreed of trees,” CORE Discussion Paper 9250, Université Catholique de Louvain (Louvain-la-Neuve, Belgium, 1992).
E.H. Aghezzaf and L.A. Wolsey, “Modelling piecewise linear concave costs in a tree partitioning problem,”Discrete Applied Mathematics 50 (1994) 101–109.
A. Balakrishnan, T.L. Magnanti and R.T. Wong, “A decomposition algorithm for expanding local access telecommunications networks,”Operations Research 43 (1995) 58–76.
I. Barany, J. Edmonds and L.A. Wolsey, “Packing and covering a tree by subtrees,”Combinatorica 6 (1986) 245–257.
E.A. Boyd, “Polyhedral results for the precedence constrained knapsack problem,”Proceedings of IPCO1 (Waterloo University Press, 1990).
C.E. Ferreira, A. Martin, C.C. de Souza, R. Weismantel and L.A. Wolsey, “The node capacitated graph partitioning problem: A computational study,” CORE Discussion Paper 9453, Université Catholique de Louvain (Louvain-la-Neuve, Belgium, 1994).
M.C. Golumbic,Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980).
H. Groeflin, T.M. Liebling and A. Prodon, “Optimal subtrees and extensions,”Annals of Discrete Mathematics 16 (1982) 121–127.
D.S. Johnson and K.A. Niemy, “On knapsacks, partitions and a new dynamic programming technique for trees,”Mathematics of Operations Research 8 (1983) 1–14.
J.A. Lukes, “Efficient algorithm for the partitioning of trees,”IBM Journal of Research and Development 18 (1974) 217–224.
P.B. Mirchandani and R.L. Francis,Discrete Location Theory (Wiley, New York, 1990).
G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley, New York, 1988).
J.E. Ward, R.T. Wong, P. Lemke and A. Oudjit, “Properties of the tree K-median linear programming relaxation,” Research Report CC-878-29, Institute for Interdisciplinary Engineering Studies, Purdue University (1987).
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Research supported in part by Nato Collaborative Research Grant CRG 900281, Science Program SC1-CT91-620 of the EEC, and contract No 26 of the programme “Pôle d'attraction interuniversitaire” of the Belgian government.
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Aghezzaf, E.H., Magnanti, T.L. & Wolsey, L.A. Optimizing constrained subtrees of trees. Mathematical Programming 71, 113–126 (1995). https://doi.org/10.1007/BF01585993
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DOI: https://doi.org/10.1007/BF01585993