Subexponential LPs Approximate Max-Cut - Hopkins, Samuel B.; Schramm, Tselil; Trevisan, Luca (Proceedings / annual Symposium on Foundations of Computer Science, 2020)
We show that for every ε > 0 , the degree -n^{ε} Sherali-Adams linear program (with \exp(\tilde{O}(n^{ε}) ) variables and constraints) approximates the maximum cut problem within a factor of ( \frac{1}{2}+ε^{′} ), for some ε^{′}(ε) > 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 \frac{1}{2} (up to the function ε^{′}(ε) ). Previously, only semidefinite programs and spectral methods were known to yield approximation factors better than \frac{1}{2} for Max-Cut in time 2^{o(n)} . We also show that constant-degree Sherali-Adams linear programs (with \text{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 [3] that ( \frac{1}{2}+ε ) approximation of Max Cut requires Ω_{ε}(n) rounds in the Lovász-Schrijver hierarchy. We also provide a subexponential time approximation for Khot's Unique Games problem [4]: 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 [5]-[7].