A simple parallel algorithm for the maximal independent set problem
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
On the independence number of random graphs
Discrete Mathematics
Locality in distributed graph algorithms
SIAM Journal on Computing
On the independence and chromatic numbers of random regular graphs
Journal of Combinatorial Theory Series B
Limits of dense graph sequences
Journal of Combinatorial Theory Series B
Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms
Theoretical Computer Science
Maximum matching in sparse random graphs
SFCS '81 Proceedings of the 22nd Annual Symposium on Foundations of Computer Science
Constant-Time Approximation Algorithms via Local Improvements
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Information, Physics, and Computation
Information, Physics, and Computation
Local Graph Partitions for Approximation and Testing
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Proceedings of the forty-second ACM symposium on Theory of computing
On the solution-space geometry of random constraint satisfaction problems
Random Structures & Algorithms
Perfect matchings as IID factors on non-amenable groups
European Journal of Combinatorics
On independent sets in random graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On belief propagation guided decimation for random k-SAT
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞. 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 nearly empty 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, such as for example, the Belief Propagation algorithm, our result is the first one where the clustering property is used to formally prove limits on local algorithms.