Covering Relaxation for Positive 0-1 Polynomial Programs

Granot and Hammer (Granot, F., P. L. Hammer. 1971. On the use of boolean functions in 0-1 programming. Operations Research, Statistics and Economic Mimeograph Series No. 70, Technion (August 1970). (Published in Methods of Oper. Res. 12 154–184); Granot, F., P. L. Hammer. 1975. On the role of generalized covering problems. Cahiers du Centre D'Etudes de Recherche Operationelle 17 277–289.) have shown constructively that the set of 0-1 vectors x satisfying a polynomial constraint with non negative coefficients remains unchanged if the polynomial constraint is replaced by an appropriate finite collection of linear inequalities in x. When rewritten in the complementary vector x̄ = 1 − x, the linear inequalities are those of a linear covering problem. This reduction is used here to give a cutting-plane algorithm for solving the positive 0-1 polynomial program of finding a 0-1 vector that maximizes a nondecreasing linear function subject to the restrictions that several polynomials do not exceed given numbers. The algorithm consists of solving (by implicit enumeration) a nested sequence of linear covering problems, each of which is a relaxation of the original positive 0-1 polynomial program. Each covering problem is found by adding to its predecessor a small number (one suffices, but more speed convergence) of covering constraints that are violated by the optimal solution found for the preceding covering problem. Over 200 randomly generated problems with up to 50 variables and 50 constraints were solved using our covering relaxation algorithm. The CPU time was often less than one second and has exceeded one minute only for three problems. The number of iterations, i.e., the number of linear covering problems solved, ranged between 1 and 6, while the number of covers in the largest covering problem solved usually did not exceed the number of constraints in the original problem.