A PTAS for weight constrained Steiner trees in series-parallel graphs
Theoretical Computer Science
Strong lower bounds for the prize collecting Steiner problem in graphs
Discrete Applied Mathematics - Brazilian symposium on graphs, algorithms and combinatorics
Primal-dual approximation algorithms for the Prize-Collecting Steiner Tree Problem
Information Processing Letters
On the complexity of the robust spanning tree problem with interval data
Operations Research Letters
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Given a graph G = (V,E) with a cost on each edge in E and a prize at each vertex in V, and a target set V′ ⊆ V, the Prize Collecting Steiner Tree (PCST) problem is to find a tree T interconnecting vertices in V′ that has minimum total costs on edges and maximum total prizes at vertices in T. This problem is NP-hard in general, and it is polynomial-time solvable when graphs G are restricted to 2-trees. In this paper, we study how to deal with PCST problem with uncertain costs and prizes. We assume that edge e could be included in T by paying cost xe ∈ [ce-, ce+] while taking risk ce+ - xe/ce+ - ce- losing e, and vertex v could be awarded prize pv ∈ [pv-,pv+] while taking risk yv-pv- / pv+ - pv- of losing the prize. We establish two risk models for the PCST problem, one minimizing the maximum risk over edges and vertices in T and the other minimizing the sum of risks. Both models are subject to upper bounds on the budget for constructing a tree. We propose two polynomial-time algorithms for these problems on 2-trees, respectively. Our study shows that the risk models have advantages over the tradional robust optimization model, which yields NP-hard problems even if the original optimization problems are polynomial-time solvable.