Subexponential LPs Approximate Max-Cut

Samuel B. Hopkins, Tselil Schramm, Luca Trevisan

2020 (modified: 03 Nov 2022)FOCS 2020Readers: Everyone

Abstract: 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 ([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 [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 [3] that ([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]-[6]-[7].

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