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 maxcut 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.
Bibliographical noteFunding Information:
Received July 2017; revised March 2018. 1Supported in part by NSF Grant DMS-1642207 and Hong Kong Research Grants Council GRF-14302515. 2Supported in part by NSERC. MSC2010 subject classifications. Primary 60K35; secondary 05C80, 60G15, 60F10, 68W20, 82B44. Key words and phrases. Local algorithms, maximum cut problems, spin glasses.
- Local algorithms
- Maximum cut problems
- Spin glasses