Subexponential LPs Approximate Max-Cut

Abstract: We show that for every $\varepsilon > 0$, the degree-$n^\varepsilon$ Sherali-Adams linear program (with $\exp(\tilde{O}(n^\varepsilon))$ variables and constraints) approximates the maximum cut problem within a factor of $(\frac{1}{2}+\varepsilon')$, for some $\varepsilon'(\varepsilon) > 0$. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut, and hence resolves the extension complexity of approximate Max-Cut for approximation factors close to $\frac{1}{2}$ (up to the function $\varepsilon'(\varepsilon)$). Previously, only semidefinite programs and spectral methods were known to yield approximation factors better than $\frac 12$ 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 Lov\'asz-Schrijver hierarchies for approximating Max-Cut, since it is known that $(\frac{1}{2}+\varepsilon)$ approximation of Max Cut requires $\Omega_\varepsilon (n)$ rounds in the Lov\'asz-Schrijver hierarchy. We also provide a subexponential time approximation for Khot's Unique Games problem: we show that for every $\varepsilon > 0$ the degree-$(n^\varepsilon \log q)$ Sherali-Adams linear program distinguishes instances of Unique Games of value $\geq 1-\varepsilon'$ from instances of value $\leq \varepsilon'$, for some $\varepsilon'( \varepsilon) >0$, 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.