A relax-and-cut algorithm for the prize-collecting Steiner problem in graphs

Authors:
Alexandre Salles da Cunha;Abilio Lucena;Nelson Maculan;Mauricio G. C. Resende
Affiliations:
Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Brazil;Departamento de Administração, Universidade Federal do Rio de Janeiro, Brazil and Programa de Engenharia de Sistemas e Computação - COPPE, Universidade Federal do Rio de Janeir ...;Programa de Engenharia de Sistemas e Computação - COPPE, Universidade Federal do Rio de Janeiro, Brazil;AT&T Labs Research, United States
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

Given an undirected graph G with penalties associated with its vertices and costs associated with its edges, a Prize Collecting Steiner (PCS) tree is either an isolated vertex of G or else any tree of G, be it spanning or not. The weight of a PCS tree equals the sum of the costs for its edges plus the sum of the penalties for the vertices of G not spanned by the PCS tree. Accordingly, the Prize Collecting Steiner Problem in Graphs (PCSPG) is to find a PCS tree with the lowest weight. In this paper, after reformulating and re-interpreting a given PCSPG formulation, we use a Lagrangian Non Delayed Relax and Cut (NDRC) algorithm to generate primal and dual bounds to the problem. The algorithm is capable of adequately dealing with the exponentially many candidate inequalities to dualize. It incorporates ingredients such as a new PCSPG reduction test, an effective Lagrangian heuristic and a modification in the NDRC framework that allows duality gaps to be further reduced. The Lagrangian heuristic suggested here dominates their PCSPG counterparts in the literature. The NDRC PCSPG lower bounds, most of the time, nearly matched the corresponding Linear Programming relaxation bounds.