The Set-Covering Problem: A New Implicit Enumeration Algorithm

The set-covering problem is to minimize cx subject to Ax ≧ e_n over all binary n-vectors x. A is an m × n binary matrix and e_n is an n-vector of 1's. We develop a new implicit enumeration strategy to solve this problem. The branching strategy is similar to the row partitioning strategy used by other authors in the partitioning problem. Simple and sharp bounds are obtained by relaxing the constraints of the associated linear program by attaching nonnegative multipliers to them. Good multipliers are obtained by using the subgradient optimization technique. Computational experience shows that these bounds are at least one order of magnitude more efficient than the ones obtained by solving the associated linear program with the simplex method. Computational results with this new implicit enumeration algorithm are encouraging. Problems with as many as 50 constraints and 100 variables were solved in the order of 100 seconds of CPU time on an IBM 360-67.