Subexponential LPs Approximate Max-Cut

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^{\prime} ), for some \varepsilon^{\prime}(\varepsi...

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Bibliographic Details
Published in:Proceedings / annual Symposium on Foundations of Computer Science pp. 943 - 953
Main Authors: Hopkins, Samuel B., Schramm, Tselil, Trevisan, Luca
Format: Conference Proceeding
Language:English
Published: IEEE 01.11.2020
Subjects:
Approximation algorithms
Combinatorial Optimization
Correlation
Eigenvalues and eigenfunctions
Games
Global Correlation Rounding
Linear programming
Max-Cut
Optimization
Programming
Sherali-Adams
Subexponential Time Algorithms
Unique Games Conjecture
ISSN:2575-8454
Online Access:Get full text
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Summary: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^{\prime} ), for some \varepsilon^{\prime}(\varepsilon) > 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 \varepsilon^{\prime}(\varepsilon) ). 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}+\varepsilon ) approximation of Max Cut requires \Omega_{\varepsilon}(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 \varepsilon > 0 the degree-( n^{\varepsilon}\log q ) Sherali-Adams linear program distinguishes instances of Unique Games of value \geq 1-\varepsilon ^{\prime} from instances of value \leq \varepsilon^{\prime} , for some \varepsilon^{\prime}(\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 [5]-[7].
ISSN:2575-8454
DOI:10.1109/FOCS46700.2020.00092