A more rapidly mixing Markov chain for graph colorings

Dyer, Martin; Greenhill, Catherine

doi:10.1002/(sici)1098-2418(199810/12)13:3/4<285::aid-rsa6>3.0.co;2-r

Cited by 52 publications

(39 citation statements)

References 15 publications

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“…We use ρ to denote the shortest-path distance in the Markov chain, i.e., ρ(x, y) is the minimum number of transitions to go from x to y. Lemma 9 (Theorem 2.1 in [9]). Suppose there exists β < 1 such that, for all x, y with P (x, y) > 0, it holds that…”

Section: Mixing Timementioning

confidence: 99%

Sorting processes with energy-constrained comparisons ^*

Geissmann

Penna

2018

Phys. Rev. E

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We study very simple sorting algorithms based on a probabilistic comparator model. In this model, errors in comparing two elements are due to (1) the energy or effort put in the comparison and (2) the difference between the compared elements. Such algorithms repeatedly compare and swap pairs of randomly chosen elements, and they correspond to natural Markovian processes. The study of these Markov chains reveals an interesting phenomenon. Namely, in several cases, the algorithm that repeatedly compares only adjacent elements is better than the one making arbitrary comparisons: in the long-run, the former algorithm produces sequences that are "better sorted". The analysis of the underlying Markov chain poses interesting questions as the latter algorithm yields a nonreversible chain, and therefore its stationary distribution seems difficult to calculate explicitly. We nevertheless provide bounds on the stationary distributions and on the mixing time of these processes in several restrictions.

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Section: Mixing Timementioning

confidence: 99%

Sorting processes with energy-constrained comparisons ^*

Geissmann

Penna

2018

Phys. Rev. E

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“…Choosing the set S carefully can simplify considerably the arguments involved in proving rapid mixing of Markov chains by coupling. The path coupling method is described in the next lemma (taken from [8]). We use the term path to refer to a sequence of elements of Ω, which need not be a path of possible transitions in the Markov chain.…”

Section: Path Couplingmentioning

confidence: 99%

“…Then recolor v with c to give the new state. A new Markov chain of colorings, denoted by M 1 (Ω k (G)), was introduced in [8]. This new chain is irreducible for k ≥ ∆+1 and is also rapidly mixing for k ≥ 2∆.…”

Section: Path Couplingmentioning

confidence: 99%

“…In a recent paper, Bubley and Dyer [3], using a new technique called path coupling, showed how a coupling proof could be extended to the Potts model (and beyond). Subsequently, Dyer and Greenhill [8] used this technique to analyze a more rapidly mixing chain on colorings. This reduces the running time for the case k = 2∆ by an Ω * (n 2 ) factor, but still does not demonstrate rapid mixing with fewer than 2∆ colors.…”

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confidence: 99%

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On Approximately Counting Colorings of Small Degree Graphs

Bubley¹,

Dyer²,

Greenhill³

et al. 1999

SIAM J. Comput.

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We consider approximate counting of colorings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree ∆ of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by considering the simplest case of 5-coloring graphs of maximum degree 3. Our proof involves a computer-assisted proof technique to establish rapid mixing of a new "heat bath" Markov chain on colorings using the method of path coupling. We outline an extension to 7-colorings of triangle-free 4-regular graphs. Since rapid mixing implies approximate counting in polynomial time, we show in contrast that exact counting is unlikely to be possible (in polynomial time). We give a general proof that the problem of exactly counting the number of proper k-colorings of graphs with maximum degree ∆ is #P-complete whenever k ≥ 3 and ∆ ≥ 3.

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“…The method of path coupling simplifies our goal by letting us bound the mixing rate of a Markov chain by considering only a small subset of ⍀ ϫ ⍀ (see [6,9].) We use the following theorem, obtained by combining Theorems 2.1 and 2.2 in Dyer and Greenhill [9].…”

Section: Bounding the Mixing Rate Ofm Nmentioning

confidence: 99%

Random dyadic tilings of the unit square

Janson

Randall

Spencer

2002

Random Struct Algorithms

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ABSTRACT:A "dyadic rectangle" is a set of the form R ϭ [a2Ϫt ], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2 n nonoverlapping dyadic rectangles, each of area 2Ϫn , whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.

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A more rapidly mixing Markov chain for graph colorings

Cited by 52 publications

References 15 publications

Sorting processes with energy-constrained comparisons ^*

Sorting processes with energy-constrained comparisons ^*

On Approximately Counting Colorings of Small Degree Graphs

Random dyadic tilings of the unit square

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A more rapidly mixing Markov chain for graph colorings

Cited by 52 publications

References 15 publications

Sorting processes with energy-constrained comparisons *

Sorting processes with energy-constrained comparisons *

On Approximately Counting Colorings of Small Degree Graphs

Random dyadic tilings of the unit square

Contact Info

Product

Resources

About

Sorting processes with energy-constrained comparisons ^*

Sorting processes with energy-constrained comparisons ^*