Solving Two-Stage Stochastic Steiner Tree Problems by Two-Stage Branch-and-Cut

Abstract

We consider the Steiner tree problem under a two-stage stochastic model with recourse and finitely many scenarios. In this problem, edges are purchased in the first stage when only probabilistic information on the set of terminals and the future edge costs is known. In the second stage, one of the given scenarios is realized and additional edges are puchased in order to interconnect the set of (now known) terminals. The goal is to decide on the set of edges to be purchased in the first stage while minimizing the overall expected cost of the solution. We provide a new semi-directed cut-set based integer programming formulation, which is stronger than the previously known undirected model. We suggest a two-stage branch-and-cut (B&C) approach in which L-shaped and integer-L-shaped cuts are generated. In our computational study we compare the performance of two variants of our algorithm with that of a B&C algorithm for the extensive form of the deterministic equivalent (EF). We show that, as the number of scenarios increases, the new approach significantly outperforms the (EF) approach.

Decomposition methods for the two-stage stochastic Steiner tree problem

“…The objective is to select first-stage edges E 0 S ⊆ E and second-stage edges E k S ⊆ E for each k ∈ K such that the subgraph induced by E 0 S ∪ E k S , G[E 0 S ∪ E k S ], connects T k and the expected cost Our contribution For the deterministic STP a wealth of theoretical results [6,11,20,29] and empirically successful computational techniques are known [8,10,31]. However, as noted in [2,27], the generalization of results from the STP to the SSTP is not straightforward. In this article we first provide a new integer linear programming (ILP) formulation for the SSTP and show that it is the strongest (in terms of the quality of linear relaxation bounds) among existing formulations.…”

“…We compare our method with the state-of-the-art exact approach from [2,27], which employs a Benders decomposition based on two-stage branchand-cut (B&C), and a genetic algorithm from [19], introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms the alternative approaches from the literature, both in terms of computing times, and the quality of obtained solutions.…”

“…An algorithm based on fixed-parameter tractability has been introduced in [23], and a genetic algorithm in [19]. The only exact method we are aware of is a two-stage B&C approach based on Benders decomposition, which has been originally proposed in [2]. Very recently, the study of a generalization of the SSTP, namely the stochastic survivable network design problem, along with a more sophisticated implementation, has been given in [27].…”

“…Consider the following two semi-directed cut formulations, (SDC 1 ) and (SDC 2 ), studied in [2,27,41].…”

“…The problem has been introduced by Gupta et al [14] and subsequently, algorithms based on fixed-parameter tractability [23], heuristics [19], and exact methods [2,27] have been proposed. Moreover, approximation algorithms for several variants of the problem have been studied [15][16][17]36].…”