Approximate constraint satisfaction requires large LP relaxations

with Siu On Chan, James R. Lee, Prasad Raghavendra. FOCS 2013.


Invited to FOCS 2013 special issue, Accepted to JACM


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 12\frac12 and any such linear program for MAX 3-SAT has an integrality gap of 78\frac78.


  • strong relaxations
  • lower bounds
  • linear programming
  • constraint satisfaction problems