Subset Algebra Lift Operators for 0-1 Integer Programming

We extend the Sherali-Adams, Lovász-Schrijver, Balas-Ceria-Cornuéjols and Lasserre lift-and-project methods for 0-1 optimization by considering liftings to subset algebras. Our methods yield polynomialtime 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.