Approximate constraint satisfaction requires large LP relaxations

with , , . FOCS 2013, arxiv:1309.0563. pdf
Invited to FOCS 2013 special issue, Accepted to JACM.

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 exactly as powerful as programs arising from a constant number of rounds of the Sherali–Adams hierarchy.

In particular, any polynomial-sized linear program for MAXCUT has an integrality gap of $$\frac12$$ and any such linear program for MAX 3-SAT has an integrality gap of $$\frac78$$.

keywords

strong relaxations, lower bounds, linear programming, constraint satisfaction problems.