Approximate Constraint Satisfaction Requires Large LP Relaxations

TL;DR

This paper establishes that for certain constraint satisfaction problems, small linear programming relaxations are insufficient to achieve better approximations, requiring large LPs or multiple Sherali-Adams rounds.

Contribution

It proves super-polynomial lower bounds on LP relaxation sizes for approximation problems, linking LP power to Sherali-Adams hierarchy levels.

Findings

Polynomial-sized LPs match Sherali-Adams with constant rounds.

Any small LP for Max Cut has an integrality gap of 1/2.

Any small LP for Max 3-Sat has an integrality gap of 7/8.

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 Max Cut has an integrality gap of 1/2 and any such linear program for Max 3-Sat has an integrality gap of 7/8.