Limits of local algorithms over sparse random graphs

Abstract

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Recent research has shown that such algorithms show significant promise in computing structures like large independent sets in graphs locally. Indeed the promise led to a conjecture by Hatami, Lovász and Szegedy [Geom. Funct. Anal. 24 (2014) 269–296] that local algorithms defined specifically as so-called i.i.d. factors may be able to find approximately largest independent sets in random $d$-regular graphs. In this paper, we refute this conjecture and show that every independent set produced by local algorithms is multiplicative factor $1/2+1/(2\sqrt{2})$ smaller than the largest, asymptotically as $d\rightarrow\infty$. Our result is based on an important clustering phenomena predicted first in the literature on spin glasses, and recently proved rigorously for a variety of constraint satisfaction problems on random graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property that we prove and apply in this paper shows that typically every two large independent sets in a random graph either have a significant intersection, or have a very small intersection. As a result, large independent sets are clustered according to the proximity to each other. While the clustering property was postulated earlier as an obstruction for the success of local algorithms, our result is the first one where the clustering property is used to formally prove limits on local algorithms.