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 0-1 matrix, xi is a random 0-1 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 polynomial-time 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, non-linear etc) of the distribution function of xi, and can be easily embedded within a branch-and-cut framework yielding a distribution-free 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 cost-vector c thereby facilitating their application in sensitivity analysis, re-optimization and warm-starting (PSC). We give preliminary computational results to illustrate our findings on a test-bed of 40 (PSC) instances created from the OR-Lib set-covering instance scp41.
Citation
Tepper Working Paper 2007-E9, Carnegie Mellon University.