On the Connection of the Sherali-adams Closure and Border Bases

The Sherali-Adams lift-and-project hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the Sherali-Adams procedure by relating it to methods from computational algebraic geometry. Our two main results are the equivalence of the Sherali-Adams procedure to the computation of a border basis, and a refinement of the Sherali-Adams procedure that arises from this new connection. We present a modified version of the border basis algorithm to generate a hierarchy of linear programming relaxations that are stronger than those of Sherali and Adams, and over which one can still optimize in polynomial time (for a fixed number of rounds in the hierarchy). In contrast to the well-known Gröbner bases approach to integer programming, our procedure does not create primal solutions, but constitutes a novel approach of using computer-algebraic methods to produce dual bounds.