The Annals of Probability

Suboptimality of local algorithms for a class of max-cut problems

Wei-Kuo Chen, David Gamarnik, Dmitry Panchenko, and Mustazee Rahman

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Abstract

We show that in random $K$-uniform hypergraphs of constant average degree, for even $K\geq 4$, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain nontrivial interval—a phenomenon referred to as the overlap gap property—which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models and showing the overlap gap property in the latter setting.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1587-1618.

Dates
Received: July 2017
Revised: March 2018
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784027

Digital Object Identifier
doi:10.1214/18-AOP1291

Mathematical Reviews number (MathSciNet)
MR3945754

Zentralblatt MATH identifier
07067277

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C80: Random graphs [See also 60B20] 60G15: Gaussian processes 60F10: Large deviations 68W20: Randomized algorithms 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Local algorithms maximum cut problems spin glasses

Citation

Chen, Wei-Kuo; Gamarnik, David; Panchenko, Dmitry; Rahman, Mustazee. Suboptimality of local algorithms for a class of max-cut problems. Ann. Probab. 47 (2019), no. 3, 1587--1618. doi:10.1214/18-AOP1291. https://projecteuclid.org/euclid.aop/1556784027


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