Subexponential LPs Approximate Max-Cut

We show that for every epsilon > 0, the degree-n(epsilon) Sherali-Adams linear program (with exp((O) over tilde (n(epsilon))) variables and constraints) approximates the maximum cut problem within a factor of (1/2 + epsilon'), for some epsilon' ( epsilon) > 0. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut [1], [2], and hence resolves the extension complexity of approximate Max-Cut for approximation factors close to 1 2 (up to the function e (e)). Previously, only semidefinite programs and spectral methods were known to yield approximation factors better than 1 2 for Max-Cut in time 2(o(n)). We also show that constant-degree Sherali-Adams linear programs (with poly(n) variables and constraints) can solve Max-Cut with approximation factor close to 1 on graphs of small threshold rank: this is the first connection of which we are aware between threshold rank and linear programming-based algorithms. Our results separate the power of Sherali-Adams versus Lovasz-Schrijver hierarchies for approximating Max-Cut, since it is known [3] that ( 1 2 + e) approximation of Max Cut requires Oe(n) rounds in the Lovasz-Schrijver hierarchy. We also provide a subexponential time approximation for Khot's Unique Games problem [4]: we show that for every epsilon > 0 the degree-(ne log q) Sherali-Adams linear program distinguishes instances of Unique Games of value = 1 - epsilon' from instances of value = epsilon', for some epsilon' (e) > 0, where q is the alphabet size. Such guarantees are qualitatively similar to those of previous subexponential-time algorithms for Unique Games but our algorithm does not rely on semidefinite programming or subspace enumeration techniques [5], [6], [7].