# Rapid Mixing for Colorings via Spectral Independence

@inproceedings{Chen2021RapidMF,
title={Rapid Mixing for Colorings via Spectral Independence},
author={Zongchen Chen and Andreas Galanis and Daniel Stefankovic and Eric Vigoda},
booktitle={SODA},
year={2021}
}
The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems. Let $\alpha^*\approx 1.763$ denote the solution to $\exp(1/x)=x$ and let $\alpha>\alpha^*$. We prove…

## Topics from this paper

Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region
• Computer Science, Mathematics
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
• 2022
The main contribution is to show how to elevate probabilistic/analytic bounds on the marginal probabilities for the typical structure of phases on random bipartite regular graphs into efficient algorithms, using the polymer method.
A Matrix Trickle-Down Theorem on Simplicial Complexes and Applications to Sampling Colorings
• Computer Science, Mathematics
ArXiv
• 2021
A matrix trickle-down theorem is established, generalizing Oppenheim’s influential result, as a new technique to prove that a high dimensional simplicial complex is a local spectral expander.
Rapid mixing of the hardcore Glauber dynamics and other Markov chains in bounded-treewidth graphs
• Computer Science
ArXiv
• 2021
A new rapid mixing result for a natural random walk on the independent sets of an input graph G is given, and the Glauber dynamics on the partial q-colorings of G mix rapidly for all λ > 0 when q ≥ ∆ + 2 is bounded.
From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk
• Kuikui Liu
• Computer Science, Physics
APPROX-RANDOM
• 2021
We show that the existence of a “good” coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More
Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion
• Computer Science, Physics
STOC
• 2021
An optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings is proved and the approximate tensorization of entropy can be deduced from entropy factorization into blocks of fixed linear size.
Spectral Independence via Stability and Applications to Holant-Type Problems
• Computer Science, Physics
ArXiv
• 2021
It is proved that if a (multivariate) partition function is nonzero in a region around a real point λ then spectral independence holds at λ, and for Holant-type problems on bounded-degree graphs, optimal O(n logn) mixing time bounds are obtained for the single-site update Markov chain known as the Glauber dynamics.
Entropic Independence in High-Dimensional Expanders: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Polynomials and the Ising Model
• Computer Science
ArXiv
• 2021
This work shows that μ is entropically independent exactly when a transformed version of the generating polynomial of μ can be upper bounded by its linear tangent, a property implied by concavity of the said transformation.
Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models
• Nima Anari, Vishesh Jain
• Computer Science, Physics
• 2021
This work establishes the tight mixing time of O(n log n) for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than 1, improving upon the prior quadratic dependence on n.
High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games
• Computer Science, Mathematics
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
• 2022
A combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders, which offers a broad generalization of the well-studied Johnson and Grassmann graphs, and a novel spectral complexity measure called Stripped Threshold Rank, which can replace the (much larger) threshold rank as a parameter controlling the performance of algorithms on structured objects.
From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization
• Computer Science, Mathematics
• 2021
This work establishes the connection between sampling and optimization by showing that an exchange inequality, a concept rooted in discrete convex analysis, can be derived from fast mixing of local random walks, and connects exchange inequalities with composable core-sets for optimization.

## References

SHOWING 1-10 OF 35 REFERENCES
Strong Spatial Mixing with Fewer Colors for Lattice Graphs
• Mathematics, Computer Science
SIAM J. Comput.
• 2005
The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence, which gives an analysis with a horizon of more than one level of induction, which leads to improved results.
Improved Bounds for Randomly Sampling Colorings via Linear Programming
• Computer Science, Physics
SODA
• 2019
Two approaches are used to give two proofs that the Glauber dynamics is rapidly mixing for any $k\ge\left(\frac{11}{6} - \epsilon_0\right)\Delta$ for some absolute constant $k > 2 \Delta$.
A Deterministic Algorithm for Counting Colorings with 2-Delta Colors
• Computer Science, Mathematics
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
• 2019
This work gives a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of q-colorings of a graph of maximum degree Delta, provided only that q ≥ 2Delta, and matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo.
Correlation decay and deterministic FPTAS for counting colorings of a graph
• Computer Science, Mathematics
J. Discrete Algorithms
• 2012
The main distinction of this work is that the correlation decay property on a computation tree arising from a certain recursive procedure is established, rather than reducing the problem to the one on a self-avoiding tree of a graph, as is done in Weitz (2006) [25].
A deterministic algorithm for counting colorings with 2Δ colors
• Computer Science
ArXiv
• 2019
A polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of q-colorings of a graph of maximum degree Δ, provided only that q ≥ 2Δ, and matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo.
Strong spatial mixing of list coloring of graphs
• Mathematics, Computer Science
Random Struct. Algorithms
• 2015
Strong spatial mixing is established for a more general problem, namely list coloring, for arbitrary bounded degree triangle-free graphs by proving the decay of correlations of marginal probabilities associated with graph nodes measured using a suitably chosen error function.
A non-Markovian coupling for randomly sampling colorings
• Mathematics, Computer Science
44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
• 2003
It is proved the Glauber dynamics is close to the uniform distribution after O(n log n) steps whenever k > (1 + /spl epsiv/)/spl Delta/, for all /Spl epsIV/ > 0.
Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
• Mathematics, Computer Science
Combinatorics, Probability and Computing
• 2016
The complementary result for the antiferromagnetic Ising model without external field is proved, namely, that unless RP = NP, for all Δ ⩾ 3, there is no FPRAS for approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree.
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
• Mathematics, Computer Science
SODA '07
• 2007
The principle insight of the present work is that the correlation decayproperty can be established with respect to certain computation tree, as opposed to the conventional correlation decay property which is typically established withrespect to graph theoretic neighborhoods of a given node.
Improved FPTAS for Multi-spin Systems
• Mathematics, Computer Science
APPROX-RANDOM
• 2013
The deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function for a class of multi-spin systems is designed and an FPTAS for the Potts models with inverse temperature β up to a critical threshold is given, confirming a conjecture in [10].