https://hal.inria.fr/hal-00794824Alvarez-Miranda, E.E.Alvarez-MirandaIndustrial Management Department [Talca] - Universidad de TalcaCandia, A.A.CandiaChen, X.X.ChenHu, X.X.HuLi, B.B.LiEfficient Algorithms for the Prize Collecting Steiner Tree Problems with Interval DataHAL CCSD2010[INFO.INFO-MC] Computer Science [cs]/Mobile ComputingMonteil, Alain2013-02-26 15:37:412022-02-08 14:16:122013-02-26 16:49:51enConference papers1Given 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 $V\'\\subseteq V$, 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 $x_eın[c_e^-,c_e^+]$ while taking risk $\\frac c_e^+-x_e{ c_e^+-c_e^-$ of losing $e$, and vertex $v$ could be awarded prize $p_vın [p_v^-,p_v^+]$ 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.