Strong lower bounds for the prize collecting Steiner problem in graphs

A. Lucena and M.G.C. Resende 

Discrete Applied Mathematics, vol. 141, pp. 277-294, 2004.


Given an undirected graph G with nonnegative edges costsand nonnegative vertex penalties, the prize collecting Steinerproblem 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 ofthe penalties of those vertices not spanned by the tree.In thispaper, we present an integer programming formulation of the PCSPGand describe an algorithm to obtain lower bounds for the problem.The algorithm is based on polyhedral cutting planes and is initiatedwith tests that attempt to reduce the size of the input graph.Computational experiments were carried out to evaluate the strengthof the formulation through its linear programming relaxation.The algorithm found optimal solutions for 99 out of the 114instances tested.On 96 instances, integer solutions were found(thus generating optimal PCSPG solutions).On all but seveninstances, lower bounds were equal to best known upper bounds (thusproving optimality of the upper bounds).Of these seven instances,four lower bounds were off by 1 of the best known upper bound.Nine new best known upper bounds were produced for the test set.

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