## The Annals of Probability

### Local algorithms for independent sets are half-optimal

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1543-1577.

Dates
Revised: June 2015
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835225

Digital Object Identifier
doi:10.1214/16-AOP1094

Mathematical Reviews number (MathSciNet)
MR3650409

Zentralblatt MATH identifier
1377.60049

#### Citation

Rahman, Mustazee; Virág, Bálint. Local algorithms for independent sets are half-optimal. Ann. Probab. 45 (2017), no. 3, 1543--1577. doi:10.1214/16-AOP1094. https://projecteuclid.org/euclid.aop/1494835225

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