Subexponential LPs Approximate Max-Cut

We show that for every ε > 0, the degree-n ε Sherali-Adams linear program (with exp( Õ(n ε )) variables and constraints) approximates the maximum cut problem within a factor of ( 12 + ε ′ ), for some ε ′ (ε) > 0. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut [CMM09, KMR17], and hence resolves the extension complexity of approximate Max-Cut for approximation factors close to 1 2 (up to the function ε ′ (ε)). 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 Lovász-Schrijver hierarchies for approximating Max-Cut, since it is known [STT07] that ( 1 2 + ε) approximation of Max Cut requires Ωε(n) rounds in the Lovász-Schrijver hierarchy.

We also provide a subexponential time approximation for Khot's Unique Games problem [Kho02]: we show that for every ε > 0 the degree-(n ε log q) Sherali-Adams linear program distinguishes instances of Unique Games of value ≥ 1 -ε ′ from instances of value ≤ ε ′ , for some ε ′ (ε) > 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 [ABS15, BRS11, GS11].