Efficient Algorithms for the Prize Collecting Steiner Tree Problems with Interval Data

Abstract

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 $x_e\in[c_e^-,c_e^+]$ while taking risk $\frac{ c_e^+-x_e}{ c_e^+-c_e^-}$ of losing e, and vertex v could be awarded prize $p_v\in [p_v^-,p_v^+]$ while taking risk $\frac{ y_v-p_v^-}{p_v^+-p_v^-}$ 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.

Supported in part by NNSF of China under Grant No. 10531070, 10771209, 10721101,10928102 and Chinese Academy of Sciences under Grant No. kjcx-yw-s7. P ⁴ Project Grant Center for Research and Applications in Plasma Physics and Pulsed Power Technology, PBCT-Chile-ACT 26, CONICYT; and Dirección de Programas de Investigación, Universidad de Talca, Chile.