Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs

Bhandari, Siddharth; Chakraborty, Sayantan

doi:10.1137/20m1366666

Cited by 3 publications

(48 citation statements)

References 21 publications

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“…As our main result, we obtain a CFTP based perfect sampler for k > (8/3 + o(1))∆. Pleasantly, our sampler has the same expected running time as in [2].…”

Section: Our Resultsmentioning

confidence: 61%

“…Recently, Chen, Delcourt, Moitra, Perarnau, and Postle [4] sharpened Vigoda's analysis to further relax the lower bound to k > (11/6 − ǫ 0 )∆, where ǫ 0 is a small absolute constant (∼ 10 −4 ). Under additional assumptions on the degree and girth of G, even less restrictive lower bounds on k are known (see, e.g., the references in [5,4,2]). The problem of efficiently (i.e.…”

Section: Sampling K-colorings: Approximately and Perfectlymentioning

confidence: 99%

“…For general graphs, where ∆ (or k) is allowed to grow with n, the only improvement of Huber's result is the recent work of Bhandari and Chakraborty [2], who provided an efficient CFTP based perfect sampler for k > 3∆. The natural question left open by their work is whether one can devise efficient perfect samplers for k < 3∆ -indeed, the sampler in [2] may be viewed as implementing a two-stage process, with natural barriers at k = 3∆ encountered (for different reasons) at both the stages (see Section 3.3 for a quick overview and [2, Section 3] for a more detailed explanation).…”

Section: Sampling K-colorings: Approximately and Perfectlymentioning

confidence: 99%

“…In particular, the standard Lemma 2.1 reduces the proof of Theorem 1.1 to the construction of a certain procedure which we call SamplerUnit. In Section 3, we provide an overview of this procedure -Section 3.1 contains some notation used throughout the paper, Section 3.2 contains a preliminary routine used by our algorithm, whose proof is presented in Appendix A, Section 3.3 provides a quick introduction to the sampler in [2], Section 3.4 provides a description of SamplerUnit, modulo the details of some primitive routines, and finally, Section 3.5 provides a high-level discussion of the key ideas underpinning the construction and analysis of our sampler. Section 4 presents and analyses our main primitivescompress, seeding, and disjoint, Section 5 completes the analysis of SamplerUpdate, and Section 6 concludes with some final remarks (including a brief sketch of how to relax the lower bound in Theorem 1.1 to (8/3 − ǫ 0 )∆) and directions for future research.…”

Section: Organizationmentioning

confidence: 99%

“…As in [8,2], our perfect sampler is based on coupling from the past (CFTP), which is a general procedure due to Propp and Wilson [14] for sampling exactly from the stationary distribution of a Markov chain. The basic idea behind CFTP is that for an ergodic Markov chain started at time −∞, its location at time 0 should be distributed according to the stationary distribution; hence, if we could determine the location at time 0 by only looking at the randomness generating the chain in the recent past, then we would have an efficient way of obtaining a sample from the stationary distribution of the chain.…”

Section: Coupling From the Pastmentioning

confidence: 99%

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Perfectly Sampling $k\geq (8/3 +o(1))Δ$-Colorings in Graphs

Jain,

Sah,

Sawhney

2020

Preprint

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We present a randomized algorithm which takes as input an undirected graph G on n vertices with maximum degree ∆, and a number of colors k ≥ (8/3 + o ∆ (1))∆, and returns -in expected time Õ(n∆ 2 log k) -a proper k-coloring of G distributed perfectly uniformly on the set of all proper k-colorings of G. Notably, our sampler breaks the barrier at k = 3∆ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also sketch how to modify our methods to relax the restriction on k to k ≥ (8/3 − ǫ 0 )∆ for an absolute constant ǫ 0 > 0.As in the work of Bhandari and Chakraborty, and the pioneering work of Huber [STOC 1998], our sampler is based on Coupling from the Past [Propp&Wilson, Random Struct. Algorithms, 1995] and the bounding chain method [Huber, STOC 1998; Häggström& Nelander, Scand. J. Statist., 1999]. Our innovations include a novel bounding chain routine inspired by Jerrum's analysis of the Glauber dynamics [Random Struct. Algorithms, 1995], as well as a preconditioning routine for bounding chains which uses the algorithmic Lovász Local Lemma [Moser&Tardos, J.ACM, 2010].

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“…As our main result, we obtain a CFTP based perfect sampler for k > (8/3 + o(1))∆. Pleasantly, our sampler has the same expected running time as in [2].…”

Section: Our Resultsmentioning

confidence: 61%

Section: Sampling K-colorings: Approximately and Perfectlymentioning

confidence: 99%

Section: Sampling K-colorings: Approximately and Perfectlymentioning

confidence: 99%

Section: Organizationmentioning

confidence: 99%

Section: Coupling From the Pastmentioning

confidence: 99%

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Perfectly Sampling $k\geq (8/3 +o(1))Δ$-Colorings in Graphs

Jain,

Sah,

Sawhney

2020

Preprint

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Perfect sampling from spatial mixing

Feng

Guo

Yin

2022

Random Struct Algorithms

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We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with subexponential neighborhood growth like ℤd, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer‐dimer models in such graphs.

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Perfectly sampling k ≥ (8/3 + o (1))Δ-colorings in graphs

Jain

Sah

Sawhney

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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We present a randomized algorithm which takes as input an undirected graph 𝐺 on 𝑛 vertices with maximum degree Δ, and a number of colors 𝑘 ≥ (8/3 + 𝑜 Δ (1))Δ, and returns -in expected time Õ (𝑛Δ 2 log 𝑘) -a proper 𝑘-coloring of 𝐺 distributed perfectly uniformly on the set of all proper 𝑘-colorings of 𝐺. Notably, our sampler breaks the barrier at 𝑘 = 3Δ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also discuss how our methods may be modified to relax the restriction on 𝑘 to 𝑘 ≥ (8/3 − 𝜖 0 )Δ for an absolute constant 𝜖 0 > 0.As in the work of Bhandari and Chakraborty, and the pioneering work of Huber [STOC 1998], our sampler is based on Coupling from the Past [Propp&Wilson, Random Struct. Algorithms, 1995] and the bounding chain method [Huber, STOC 1998; Häggström& Nelander, Scand. J. Statist., 1999]. Our innovations include a novel bounding chain routine inspired by Jerrum's analysis of the Glauber dynamics [Random Struct. Algorithms, 1995], as well as a preconditioning routine for bounding chains which uses the algorithmic Lovász Local Lemma [Moser&Tardos, J.ACM, 2010]. CCS CONCEPTS• Mathematics of computing → Graph coloring.

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Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs

Cited by 3 publications

References 21 publications

Perfectly Sampling $k\geq (8/3 +o(1))Δ$-Colorings in Graphs

Perfectly Sampling $k\geq (8/3 +o(1))Δ$-Colorings in Graphs

Perfect sampling from spatial mixing

Perfectly sampling k ≥ (8/3 + o (1))Δ-colorings in graphs

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