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. 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.

References

1. D. Achlioptas, A. Coja-Oghlan, and F. Ricci-Tersenghi. On the solution space geometry of random formulas. Random Structures and Algorithms, 38:251--268, 2011.
2. D. Aldous. Some open problems. phhttp://www.stat.berkeley.edu/~aldous/ Research/OP/index.html.
3. N. Alon and J. Spencer. Probabilistic Method. Wiley, 1992.
4. M. Bayati, D. Gamarnik, and P. Tetali. Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Annals of Probability, to appear. Conference version in Proc. 42nd Ann. Symposium on the Theory of Computing (STOC), 2010.
5. C. Borgs, J. Chayes, and D. Gamarnik. Convergent sequences of sparse graphs: A large deviations approach. arxiv:1302.4615 {math.PR}, 2013.
6. C. Borgs, J. Chayes, L. Lovász, and J. Kahn. Left and right convergence of graphs with bounded degree. http://arxiv.org/abs/1002.0115, 2010.
7. C. Borgs, J. Chayes, L. Lovász, V. Sós, and K. Vesztergombi. Convergent graph sequences II: Multiway cuts and statistical physics. Ann. of Math., 176:151--219, 2012.
8. C. Borgs, J. Chayes, L. Lovász, V. Sós, and K. Vesztergombi. Convergent graph sequences I: Subgraph frequencies, metric properties, and testing. Advances in Math., 219:1801--1851, 208.
9. A. Coja-Oghlan. On belief propagation guided decimation for random k-sat. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 957--966. SIAM, 2011.
10. A. Coja-Oghlan and C. Efthymiou. On independent sets in random graphs. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 136--144. SIAM, 2011.
11. E. Csoka and G. Lippner. Invariant random matchings in cayley graphs. arXiv preprint arXiv:1211.2374, 2012.
12. G. Elek and G. Lippner. Borel oracles. an analytical approach to constant-time algorithms. In Proc. Amer. Math. Soc, volume 138, pages 2939--2947, 2010.
13. A. Frieze. On the independence number of random graphs. Discrete Mathematics, 81:171--175, 1990.
14. A. Frieze and T. Łuczak. On the independence and chromatic numbers of random regular graphs. Journal of Combinatorial Theory, Series B, 54(1):123--132, 1992.
15. D. Gamarnik and M. Sudan. Limits of local algorithms over sparse random graphs. Electronic Colloquium on Computational Complexity (ECCC), 20:55, 2013.
16. A. Hassidim, J. A. Kelner, H. N. Nguyen, and K. Onak. Local graph partitions for approximation and testing. In FOCS, pages 22--31. IEEE Computer Society, 2009.
17. H. Hatami, L. Lovász, and B. Szegedy. Limits of local-global convergent graph sequences. Preprint at http://arxiv.org/abs/1205.4356, 2012.
18. R. Karp and M. Sipser. Maximum matchings in sparse random graphs. In 22nd Annual Symposium on Foundations of Computer Science, pages 364--375, 1981.
19. N. Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193--201, 1992.
20. L. Lovász and B. Szegedy. Limits of dense graph sequences. Journal of Combinatorial Theory, Series B, 96:933--957, 2006.
21. M. Luby. A simple parallel algorithm for the maximal independent set problem. In R. Sedgewick, editor, STOC, pages 1--10. ACM, 1985.
22. R. Lyons and F. Nazarov. Perfect matchings as iid factors on non-amenable groups. European Journal of Combinatorics, 32(7):1115--1125, 2011.
23. M. Mezard and A. Montanari. Information, Physics and Computation. Oxford graduate texts, 2009.
24. M. Mézard, T. Mora, and R. Zecchina. Clustering of solutions in the random satisfiability problem. Physical Review Letters, 94(19):197205, 2005.
25. H. N. Nguyen and K. Onak. Constant-time approximation algorithms via local improvements. In FOCS, pages 327--336. IEEE Computer Society, 2008.
26. M. Parnas and D. Ron. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theor. Comput. Sci., 381(1-3):183--196, 2007.
27. R. Rubinfeld, G. Tamir, S. Vardi, and N. Xie. Fast local computation algorithms. In B. Chazelle, editor, ICS, pages 223--238. Tsinghua University Press, 2011.
28. M. Talagrand. Mean Field Models for Spin Glasses: Volume I: Basic Examples. Springer, 2010.
29. M. J. Wainwright and M. I. Jordan. A variational principle for graphical models. In New Directions in Statistical Signal Processing: From Systems to Brain. Cambridge, MA: MIT Press, 2005.

Index Terms

• Theory of computation

  • Computational complexity and cryptography

    • Complexity classes