Random generation of combinatorial structures from a uniform
Theoretical Computer Science
1-factorizations of random regular graphs
Random Structures & Algorithms
The complexity of counting colourings and independent sets in sparse graphs and hypergraphs
Computational Complexity
On Counting Independent Sets in Sparse Graphs
SIAM Journal on Computing
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Adaptive simulated annealing: A near-optimal connection between sampling and counting
Journal of the ACM (JACM)
Computational Transition at the Uniqueness Threshold
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Guest column: complexity dichotomies of counting problems
ACM SIGACT News
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions
SIAM Journal on Computing
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We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph G = (V,E) and an activity λ 0, we are interested in the quantity ZG(λ) defined as the sum over independent sets I weighted as w(I) = λ|I|. In statistical physics, ZG(λ) is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negibile size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and λ c(TΔ) := (Δ - 1)Δ-1/(Δ - 2)Δ. The quantity λc(TΔ) is the critical point for the so-called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for λc(TΔ) c(TΔ) + ε(Δ), unless NP=RP, for some function ε(Δ) 0. We remove the upper bound in the assumptions of Sly's result for Δ ≠ 4, 5, that is, we show that there does not exist efficient randomized approximation algorithms for all λ λc(TΔ) for Δ = 3 and Δ ≠ 6. Sly's inapproximability result uses a clever reduction, combined with a second-moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs.