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Anureet Saxena (anureetsandrew.cmu.edu) Abstract: In this paper we address the following probabilistic version (PSC) of the set covering problem: $min \{ cx \ \ {\mathbb P} (Ax\ge \xi) \ge p,\ x_{j}\in \{0,1\}^N\}$ where $A$ is a 01 matrix, $\xi$ is a random 01 vector and $p\in (0,1]$ is the threshold probability level. We formulate (PSC) as a mixed integer nonlinear program (MINLP) and linearize the resulting (MINLP) to obtain a MIP reformulation. We introduce the concepts of pinefficiency and polarity cuts. While the former is aimed at reducing the number of constraints in our model, the later is used as a strengthening device to obtain stronger formulations. A hierarchy of relaxations for (PSC) is introduced, and fundamental relationships between the relaxations are established culminating with a MIP reformulation of (PSC) with no additional integer constrained variables. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of $\xi$ are pairwise independent, the distribution function of $\xi$ is a stationary distribution or has the socalled disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a testbed consisting of almost 10,000 probabilistic instances. This testbed was created using deterministic instances from the literature and consists of probabilistic variants of the setcovering model and capacitated versions of facility location, warehouse location and kmedian models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to fraction of seconds. Keywords: Probabilistic Programming, Set Covering, Mixed Integer Programming, Cutting Planes Category 1: Stochastic Programming Category 2: Integer Programming (01 Programming ) Category 3: Integer Programming (Cutting Plane Approaches ) Citation: To appear in Mathematical Programming. Download: Entry Submitted: 02/06/2007 Modify/Update this entry  
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