Approximate Constraint Satisfaction Requires Large LP Relaxations

Linear programming is a very powerful tool for attacking optimization problems. Techniques such as the ellipsoid method have shown that linear programs are solvable in polynomial time. Furthermore, it is known linear programming is P-complete. Therefore, if one was to show that some NP-hard problem admitted a polynomial-size linear program, then P = NP. In an attempt to rule out this approach, Yannakakis [4] gave a framework for proving lower bounds on a large class of linear programs known as extended formulations. Consider the 3XORn problem on n variables. It’s not NP-hard, but it will serve as a good running example. An instance of Π ∈ 3XORn consists of m parity constraints {P1, . . . , Pm}, P` : {±1} → {0, 1} where P`(x) := xi ⊕ xj ⊕ xk = a`, for i, j, k ∈ [n] and a` ∈ {±1};