Algorithms for large scale set covering problems

This paper is concerned with the set covering problem (SCP), and in particular with the development of a new algorithm capable of solving large-scale SCPs of the size found in real-life situations. The set covering problem has a wide variety of practical applications which, in general, yield large and sparse instances, normally with hundreds of rows and thousands of columns. In this paper, we present an algorithm capable of solving problems of this size and test problems up to 400 rows and 4000 columns are solved. The method developed in this paper consists of a tree-search procedure based on a combination of decomposition and state space relaxation which is a technique developed for obtaining lower bounds on the dynamic program associated with a combinatorial optimization problem. The large size SCPs are decomposed into many smaller SCPs which are then solved or bounded by state space relaxation (SSR). Before using the decomposition and SSR, reductions both in the number of columns and the number of rows of the problem are made by applying Lagrangian relaxation, linear programming and heuristic methods.