Open Access
May 2017 Local algorithms for independent sets are half-optimal
Mustazee Rahman, Bálint Virág
Ann. Probab. 45(3): 1543-1577 (May 2017). DOI: 10.1214/16-AOP1094


We show that the largest density of factor of i.i.d. independent sets in the $d$-regular tree is asymptotically at most $(\log d)/d$ as $d\to\infty$. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random $d$-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically $2(\log d)/d$. We prove analogous results for Poisson–Galton–Watson trees, which yield bounds for local algorithms on sparse Erdős–Rényi graphs.


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Mustazee Rahman. Bálint Virág. "Local algorithms for independent sets are half-optimal." Ann. Probab. 45 (3) 1543 - 1577, May 2017.


Received: 1 May 2014; Revised: 1 June 2015; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1377.60049
MathSciNet: MR3650409
Digital Object Identifier: 10.1214/16-AOP1094

Primary: 05C80 , 60G10
Secondary: 05C69 , 68W20

Keywords: Factor of i.i.d. , Independent set , Local algorithm , Random graphs

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • May 2017
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