# Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region

@inproceedings{Chen2022SamplingCA,
title={Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region},
author={Zongchen Chen and Andreas Galanis and Daniel Stefankovic and Eric Vigoda},
booktitle={SODA},
year={2022}
}
• Published in SODA 4 May 2021
• Mathematics, Computer Science
For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bounded-degree graphs. We present new algorithmic results for approximating the partition function and sampling from the Gibbs distribution for spin systems in the non-uniqueness region on random regular bipartite graphs. We give an $\mathsf{FPRAS}$ for counting…
6 Citations
Approximately counting independent sets in bipartite graphs via graph containers
• Mathematics, Computer Science
SODA
• 2022
By implementing algorithmic versions of Sapozhenko’s graph container methods, new algorithms for approximating the number of independent sets in bipartite graphs satisfying a weak expansion condition are given.
Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs
• Mathematics, Computer Science
ArXiv
• 2020
The limiting distribution of the weights of the ordered and disordered phases at criticality is determined and exponential decay of correlations away fromcriticality is proved.
Fast and perfect sampling of subgraphs and polymer systems
• Mathematics
ArXiv
• 2022
We give an eﬃcient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets ) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a
Inapproximability of counting hypergraph colourings
• Mathematics
ArXiv
• 2021
It is shown that for all even q ≥ 4 it is NP-hard to approximate the number of colourings when Δ ≳ qK/2, and a far more refined bound for the counting problem is obtained that goes well beyond the hardness of finding a colouring and which is asymptotically tight (up to constant factors).
Computational thresholds for the fixed-magnetization Ising model
• Mathematics
STOC
• 2022
It is shown that hidden inside the model are hard computational problems for approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization, which are found for the class of bounded-degree graphs.
Technical Report Column
This report presents a meta-modelling system that automates the very labor-intensive and therefore time-heavy and therefore expensive and expensive process of manually cataloging and cataloging all the components of a smart phone.

## References

SHOWING 1-10 OF 27 REFERENCES
Counting independent sets and colorings on random regular bipartite graphs
• Mathematics
APPROX-RANDOM
• 2019
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every $\Delta$-regular bipartite graph if $\Delta\ge 53$. In the weighted case, for all
Rapid Mixing for Colorings via Spectral Independence
• Mathematics
SODA
• 2021
The spectral independence approach for colorings is developed, and new algorithmic results for the corresponding counting/sampling problems are obtained.
Improved Bounds for Randomly Sampling Colorings via Linear Programming
• Mathematics
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 $k > 2 \Delta$.
Counting independent sets in unbalanced bipartite graphs
• Mathematics, Computer Science
SODA
• 2020
We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the
The Computational Hardness of Counting in Two-Spin Models on d-Regular Graphs
• Mathematics
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• 2012
The proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting free energy density'' which coincides with the (non-rigorous) Be the prediction of statistical physics.
Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
• Mathematics
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.
Rapid Mixing from Spectral Independence beyond the Boolean Domain
• Mathematics
SODA
• 2021
This work shows that Glauber dynamics for sampling proper $q$-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree $\Delta$ provided $q\ge (\alpha^*+\delta)\Delta$, and establishes spectral independence in this regime with possibly unbounded $\Delta$.
The Complexity of Choosing an H-Coloring (Nearly) Uniformly at Random
• Mathematics
SIAM J. Comput.
• 2004
It is shown that for any fixed graph H with no trivial components, there is unlikely to be any polynomial almost uniform sampler (PAUS) for H-colorings, and it is introduced the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserve reduction.
Computational Transition at the Uniqueness Threshold
• Allan Sly
• Mathematics, Computer Science
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
It is proved that at the uniqueness threshold of the hardcore model on the d-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree \$d.
Fast algorithms for general spin systems on bipartite expanders
• Mathematics, Computer Science
MFCS
• 2020
This work develops fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large, which guarantees that the spin system is in the so-called low-temperature regime.