We give an overview of new and existing cut- and flow-based ILP formulations for the two-stage stochastic Steiner tree problem and compare the strength of the LP relaxations.
Although the STP allows constant factor approximations the stochastic problems are harder to approximate. @cite_26 showed that the group Steiner tree problem, which is @math -hard to approximate, can be reduced to the stochastic shortest path problem (a special case of the (r)SSTP). Nevertheless, in literature stochastic versions of the STP have been mostly investigated for approximation algorithms. Due to the inapproximability results restricted versions have been considered to obtain approximation algorithms, e.g., by introducing a fixed and or uniform inflation factor or a global terminal (a vertex being a terminal in all scenarios). Moreover, different models of scenario representations are used. Here, we concentrate on the finite polynomial scenario model where the random variables of the stochastic problems are assumed to have finite support. Other publications consider the black box oracle model. For an overview of these concepts see, e.g., @cite_15 .
@cite_15 @cite_26
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We consider the Steiner tree problem in graphs under uncertainty, the so-called two-stage stochastic Steiner tree problem (SSTP). The problem consists of two stages: In the first stage, we do not know which nodes need to be connected. Instead, we know costs at which we may buy edges, and a set of possible scenarios one of which will arise in the second stage. Each scenario consists of its own terminal set, a probability, and second-stage edge costs. We want to find a selection of first-stage edges and second-stage edges for each scenario that minimizes the expected costs and satisfies all connectivity requirements. We show that SSTP is in the class of fixed-parameter tractable problems (FPT), parameterized by the number of terminals. Additionally, we transfer our results to the directed and the prize-collecting variant of SSTP. We study variations of Steiner tree problem. Let P = p1, p2, ...,p n be a set of n terminals in the Euclidean plane. For a positive integer k, the bottleneck Steiner tree problem (BSTP for short) is to find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. For a positive constant R, the Steiner tree problem with minimum number of Steiner points (STP - MSP for short) asks for a Steiner tree such that each edge in the tree has length at most R and the number of Steiner points is minimized. In this paper, we give (1) a ratio-√3 + e approximation algorithm for BSTP, where e is an arbitrary positive number; (2) a ratio-3 approximation algorithm for STP-MSP with running time O(n 3 ); (3) a ratio-5 2 approximation algorithm for STP-MSP. This paper considers the Steiner tree problem in the model of two-stage stochastic optimization with recourse. This model, the focus of much recent research [11, 16, 8, 18], tries to capture the fact that many infrastructure planning problems have to be solved in the presence of uncertainty, and that we have make decisions knowing merely market forecasts (and not the precise set of demands); by the time the actual demands arrive, the costs may be higher due to inflation. In the context of the Stochastic Steiner Tree problem on a graph G = (V,E), the model can be paraphrased thus: on Monday, we are given a probability distribution π on subsets of vertices, and can build some subset EM of edges. On Tuesday, a set of terminals D materializes (drawn from the same distribution π). We now have to buy edges ET so that the set EM ∪ ET forms a Steiner tree on D. The goal is to minimize the expected cost of the solution. We give the first constant-factor approximation algorithm for this problem. To the best of our knowledge, this is the first O(1)-approximation for the stochastic version of a non sub-additive problem. In fact, algorithms for the unrooted stochastic Steiner tree problem we consider are powerful enough to solve the Multicommodity Rent-or-Buy problem, itself a topic of recent interest [3, 7, 15]. Given a set of input points, the Steiner Tree Problem (STP) is to find a minimum-length tree that connects the input points, where it is possible to add new points to minimize the length of the tree. Solving the STP is of great importance since it is one of the fundamental problems in network design, very large scale integration routing, multicast routing, wire length estimation, computational biology, and many other areas. However, the STP is NP-hard, which shatters any hopes of finding a polynomial-time algorithm to solve the problem exactly. This is why the majority of research has looked at finding efficient heuristic algorithms. Additionally, many authors focused their work on utilizing the ever-increasing computational power and developed many parallel and distributed methods for solving the problem. In this way we are able to obtain better results in less time than ever before. Here, we present a survey of the parallel and distributed methods for solving the STP and discuss some of their applications.