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A-priori Upper Bounds for the Set Covering Problem

In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic results, and it depends only on the number of rows of the coefficient matrix and the row densities. We also consider the particular case of matrices that are \textit{almost} block decomposable, and show how the bound may improve according to the particular decomposition adopted. Such final result may provide interesting indications for comparing different matrix decomposition strategies.