by Abilio Lucena, Mauricio, G. C. Resende
Discrete Applied Mathematics
http://www.research.att.com/~mgcr/doc/pcspglp.pdf
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Abstract:
Abstract. Given an undirected graph G with nonnegative edges costs and nonnegative vertex penalties, the prize collecting Steiner problem in graphs (PCSPG) seeks a tree of G with minimum weight. The weight of a tree is the sum of its edge costs plus the sum of the penalties of those vertices not spanned by the tree. In this paper we present an integer programming formulation of the PCSPG and describe an algorithm to obtain lower bounds for the problem. The algorithm is based on polyhedral cutting planes and is initiated with tests that attempt to reduce the size of the input graph. Computational experiments were carried out to evaluate the strength of the formulation through its linear programming relaxation. The algorithm found optimal solutions in 85 % of 114 problems tested. Of those optimal solutions, 97 % were integral, thus producing feasible upper bounds. Nine new best known upper bounds were produced for the test set. Tight lower bounds were produced in 89 % of the instances. Where tight lower bounds were not produced, the algorithm produced bounds with at most a 1.3 % relative deviation from the best known upper bounds. The formulation is extended to encompass a wider class of problems, namely the minimum spanning tree problem, the Steiner problem in graphs, the node weighted Steiner problem in graphs, and the minimum cost tree problem. 1.
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