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

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

Abstract

We show that in random K -uniform hypergraphs of constant average degree, for even K ≥ 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.

Citation

Chen, W.-K., Gamarnik, D., Panchenko, D., & Rahman, M. (2019). Suboptimality of local algorithms for a class of max-cut problems. Annals of Probability, 47(3), 1587-1618. https://doi.org/10.1214/18-aop1291

• Journal Article Type: Article
• Acceptance Date: May 25, 2018
• Online Publication Date: May 2, 2019
• Publication Date: 2019-05
• Deposit Date: Sep 25, 2019
• Publicly Available Date: Oct 4, 2021
• Journal: Annals of Probability
• Print ISSN: 0091-1798
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 47
• Issue: 3
• Pages: 1587-1618
• DOI: https://doi.org/10.1214/18-aop1291
• Public URL: https://durham-repository.worktribe.com/output/1320316