SIAM Journal on Computing
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
Approximately counting up to four (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Simulated Annealing in Convex Bodies and an 0*(n4) Volume Algorithm
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Counting and sampling H-colourings
Information and Computation
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Randomly Coloring Constant Degree Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A second threshold for the hard-core model on a Bethe lattice
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
Combinatorial criteria for uniqueness of Gibbs measures
Random Structures & Algorithms
Random Structures & Algorithms
Simple deterministic approximation algorithms for counting matchings
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Counting good truth assignments of random k-SAT formulae
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Graphical Models, Exponential Families, and Variational Inference
Foundations and Trends® in Machine Learning
Approximating the volume of unions and intersections of high-dimensional geometric objects
Computational Geometry: Theory and Applications
Parameter testing in bounded degree graphs of subexponential growth
Random Structures & Algorithms
Hi-index | 0.00 |
We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by recent developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide ∈-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, unlike Markov chain based algorithms, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular n-node graph with large girth has asymptotically (1.494 ...)n independent sets, and in every r-regular graph with n nodes and large girth the number of q ≥ r + 1-proper colorings is asymptotically (q(1-1/q)r/2)n for large n. In statistical physics terminology, we compute explicitly the partition function (free energy) in these cases. We extend our results to random regular graphs graphs also. The explicit results obtained in this paper would be hard to derive via Markov chain sampling technique.