Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
Author(s)
Bayati, Mohsen; Gamarnik, David; Tetali, Prasad
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We establish the existence of free energy limits for several
sparse random hypergraph models corresponding to certain
combinatorial models on Erd¨os-R´enyi graph G(N, c/N) and
random r-regular graph G(N, r). For a variety of models, including
independent sets, MAX-CUT, Coloring and K-SAT,
we prove that the free energy both at a positive and zero
temperature, appropriately rescaled, converges to a limit as
the size of the underlying graph diverges to infinity. In the
zero temperature case, this is interpreted as the existence of
the scaling limit for the corresponding combinatorial optimization
problem. For example, as a special case we prove
that the size of a largest independent set in these graphs,
normalized by the number of nodes converges to a limit
w.h.p., thus resolving an open problem, (see Conjecture 2.20
in [Wor99], as well as [Ald],[BR],[JT08] and [AS03]).
Our approach is based on extending and simplifying the
interpolation method of Guerra and Toninelli. Among other
applications, this method was used to prove the existence
of free energy limits for Viana-Bray and K-SAT models on
Erd¨os-R´enyi graphs. The case of zero temperature was treated
by taking limits of positive temperature models. We provide
instead a simpler combinatorial approach and work with the
zero temperature case (optimization) directly both in the
case of Erd¨os-R´enyi graph G(N, c/N) and random regular
graph G(N, r). In addition we establish the large deviations
principle for the satisfiability property for constraint
satisfaction problems such as Coloring, K-SAT and NAEK-
SAT. For example, let p(c, q,N) and p(r, q,N) denote,
respectively, the probability that random graphs G(N, c/N)
and G(N, r) are properly q-colorable. We prove the existence
of limits of N[superscript −1] log p(c, q,N) and N[superscript 1] log p(r, q,N),
as N -> infinity.
Date issued
2010-06Department
Sloan School of ManagementJournal
Proceedings of the 42nd ACM Symposium on Theory of Computing
Publisher
Association for Computing Machinery
Citation
Bayati, Mohsen, David Gamarnik, and Prasad Tetali. “Combinatorial Approach to the Interpolation Method and Scaling Limits in Sparse Random Graphs.” Proceedings of the 42nd ACM Symposium on Theory of Computing - STOC ’10. Cambridge, Massachusetts, USA, 2010. 105.
Version: Author's final manuscript
ISBN
978-1-4503-0050-6