Subset algebra lift operators for 0-1 integer programming

We extend the Sherali-Adams, Lovasz-Schrijver, Balas-Ceria-Cornuejols, and Lasserre lift-and-project methods for 0-1 optimization by considering liftings to subset algebras. Our methods yield polynomial-time algorithms for solving a relaxation of a set-covering problem at least as strong as that given by the set of all valid inequalities with small coefficients and, more generally, all valid inequalities where the right-hand side is not very large relative to the positive coefficients in the left-hand side. Applied to generalizations of vertex-packing problems, our methods yield, in polynomial time, relaxations that have unbounded rank using, for example, the N+ operator.