Approximate Constraint Satisfaction Requires Large LP Relaxations

Abstract

We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are no more powerful than programs arising from a constant number of rounds of the Sherali--Adams hierarchy. In particular, any polynomial-sized linear program for M ax C ut has an integrality gap of ½ and any such linear program for M ax 3-S at has an integrality gap of ⅞.