Solving two-stage stochastic network design problems to optimality

The Steiner tree problem (STP) is a central and well-studied graph-theoretical combinatorial optimization problem which plays an important role in various applications. It can be stated as follows: Given a weighted graph and a set of terminal vertices, find a subset of edges which connects the terminals at minimum cost. However, in real-world applications the input data might not be given with certainty or it might depend on future decisions. For the STP, for example, edge costs representing the costs of establishing links may be subject to inflations and price deviations. In this thesis we tackle data uncertainty by using the concept of stochastic programming and we study the two-stage stochastic version of the Steiner tree problem (SSTP). Thereby, a set of scenarios defines the possible outcomes of a random variable; each scenario is given by its realization probability and defines a set of terminals and edge costs. A feasible solution consists of a subset of edges in the first stage and edge subsets for all scenarios (second stage) such that each terminal set is connected. The objective is to find a solution that minimizes the expected cost. We consider two approaches for solving the SSTP to optimality: combinatorial algorithms, in particular fixed-parameter tractable (FPT) algorithms, and methods from mathematical programming. Regarding the combinatorial algorithms we develop a linear-time algorithm for trees, an FPT algorithm parameterized by the number of terminals, and we consider treewidth-bounded graphs where we give the first FPT algorithm parameterized by the combination of treewidth and number of scenarios. The second approach is based on deriving strong integer programming (IP) formulations for the SSTP. By using orientation properties we introduce new semi-directed cut- and flow-based IP formulations which are shown to be stronger than the undirected models from the literature. To solve these models to optimality we use a decomposition-based two-stage branch&cut algorithm, which is improved by a fast and efficient method for strengthening the optimality cuts. Moreover, we develop new and stronger integer optimality cuts. The computational performance is evaluated in a comprehensive computational study, which shows the superiority of the new formulations, the benefit of the decomposition, and the advantage of using the strengthened optimality cuts. The Steiner forest problem (SFP) is a related problem where sets of terminals need to be connected. On the one hand, the SFP is a generalization of the STP and on the other hand, we show that the SFP is a special case of the SSTP. Therefore, our results are transferable to the SFP and we present the first FPT algorithm for treewidth-bounded graphs and we model new and stronger (semi-)directed cut- and flow-based IP formulations for the SFP. In the second part of this thesis we consider the two-stage stochastic survivable network design problem, an extension of the SSTP where pairs of vertices may demand a higher connectivity. Similarly to the first part we introduce new and stronger semi-directed cut-based models, apply the same decomposition along with the cut strengthening technique, and argue the validity of the newly introduced integer optimality cuts. A computational study shows the benefit, robustness, and good performance of the decomposition and the cut strengthening method

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum