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Anureet Saxena(anureetcmu.edu) Abstract: In this paper we address the following probabilistic version (PSC) of the set covering problem: min { cx  P (Ax>= xi) >= p, x_{j} in {0,1} j in N} where A is a 01 matrix, xi is a random 01 vector and p in (0,1] is the threshold probability level. In a recent development Saxena, Goyal and Lejeune proposed a MIP reformulation of (PSC) and reported extensive computational results with small and medium sized (PSC) instances. Their reformulation, however, suffers from the curse of exponentiality  the number of constraints in their model can grow exponentially rendering the MIP reformulation intractable for all practical purposes. In this paper, we give a polynomialtime algorithm to separate the (possibly exponential sized) constraint set of their MIP reformulation. Our separation routine is independent of the specific nature (concave, convex, linear, nonlinear etc) of the distribution function of xi, and can be easily embedded within a branchandcut framework yielding a distributionfree algorithm to solve (PSC). The resulting algorithm can solve (PSC) instances of arbitrarily large block sizes by generating only a small subset of constraints in the MIP reformulation and verifying the remaining constraints implicitly. Furthermore, the constraints generated by the separation routine are independent of the coefficient matrix A and costvector c thereby facilitating their application in sensitivity analysis, reoptimization and warmstarting (PSC). We give preliminary computational results to illustrate our findings on a testbed of 40 (PSC) instances created from the ORLib setcovering instance scp41. 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: Tepper Working Paper 2007E9, Carnegie Mellon University. Download: [Postscript][PDF] Entry Submitted: 03/15/2007 Modify/Update this entry  
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