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.
Several combinatorial optimization problems choose elements to minimize the total cost of constructing a feasible solution that satisfies requirements of clients. In the S teiner T ree problem, for example, edges must be chosen to connect terminals (clients); in V ertex C over , vertices must be chosen to cover edges (clients); in F acility L ocation , facilities must be chosen and demand vertices (clients) connected to these chosen facilities. We consider a stochastic version of such a problem where the solution is constructed in two stages: Before the actual requirements materialize, we can choose elements in a first stage. The actual requirements are then revealed, drawn from a pre-specified probability distribution π thereupon, some more elements may be chosen to obtain a feasible solution for the actual requirements. However, in this second (recourse) stage, choosing an element is costlier by a factor of σ> 1. The goal is to minimize the first stage cost plus the expected second stage cost.We give a general yet simple technique to adapt approximation algorithms for several deterministic problems to their stochastic versions via the following method. First stage: Draw σ independent sets of clients from the distribution π and apply the approximation algorithm to construct a feasible solution for the union of these sets. Second stage: Since the actual requirements have now been revealed, augment the first-stage solution to be feasible for these requirements. We use this framework to derive constant factor approximations for stochastic versions of V ertex C over , S teiner T ree and U ncapacitated F acility L ocation for arbitrary distributions π in one fell swoop. For special (product) distributions, we obtain additional and improved results. Our techniques adapt and use the notion of strict cost-shares introduced in [5]. 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]. We consider two- and multistage versions of stochastic combinatorial optimization problems with recourse: in this framework, the instance for the combinatorial optimization problem is drawn from a known probability distribution @math and is only revealed to the algorithm over two (or multiple) stages. At each stage, on receiving some more information about the instance, the algorithm is allowed to build some partial solution. Since the costs of elements increase with each passing stage, there is a natural tension between waiting for later stages, to gain more information about the instance, and purchasing elements in earlier stages, to take advantages of lower costs. We provide approximation algorithms for stochastic combinatorial optimization problems (such as the Steiner tree problem, the Steiner network problem, and the vertex cover problem) by means of a simple sampling-based algorithm. In every stage, our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample. We show that if one can construct cost-sharing functions associated with the algorithms used to construct these partial solutions, then this strategy results in provable approximation guarantees for the overall stochastic optimization problem. We also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem, a problem that does not fit into the simpler framework of our main model. This issue’s column is written by guest columnists, David Shmoys and Chaitanya Swamy. Iam delighted that they agreed to write this timely column on the topic related to stochasticoptimization that has received much attention recently. Their column introduces the readerto several recent results and provides references for further readings.Samir Khuller Combinatorial optimization is often used to "plan ahead," purchasing and allocating resources for demands that are not precisely known at the time of solution. This advance planning may be done because resources become very expensive to purchase or difficult to allocate at the last minute when the demands are known. In this work we study the tradeoffs involved in making some purchase allocation decisions early to reduce cost while deferring others at greater expense to take advantage of additional, late-arriving information. We consider a number of combinatorial optimization problems in which the problem instance is uncertain---modeled by a probability distribution---and in which solution elements can be purchased cheaply now or at greater expense after the distribution is sampled. We show how to approximately optimize the choice of what to purchase in advance and what to defer.