## Path coupling using stopping times and counting independent sets and colourings in hypergraphs (2005)

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Citations: | 5 - 1 self |

### BibTeX

@MISC{Bordewich05pathcoupling,

author = {Magnus Bordewich and Martin Dyer and Marek Karpinski},

title = {Path coupling using stopping times and counting independent sets and colourings in hypergraphs},

year = {2005}

}

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### Abstract

We analyse the mixing time of Markov chains using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree ∆ of a vertex and the minimum size m of an edge satisfy m ≥ 2∆+1. We also show that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m ≥ 4 and q> ∆, and if m = 3 and q ≥ 1.65∆. We give related results on the hardness of exact and approximate counting for both problems. 1

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Citation Context ...f such a threshold exists. Counting q-colourings of hypergraphs was considered by Bubley [2], who showed that the Glauber dynamics was rapidly mixing if q >= 2\Delta , generalising a result of Jerrum =-=[19]-=- and Salas and Sokal [25] for graphs. Much work has been done on improving this result for graph colourings, see [8] and its references, but little attention appears to have been given to the hypergra... |

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Citation Context ... times in conjunction with path coupling to bound the convergence of time of Markov chains. Our main interest is in applying these results to randomised approximate counting. For an introduction, see =-=[20]-=-. To illustrate our methods, we consider approximation of the numbers of independent sets and q-colourings in hypergraphs with upperbounded degree, and lower-bounded edge size. These problems in hyper... |

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Citation Context ...ill usually give a tighter bound on mixing time than [17, Corollary 3]. See Remark 2.3 below. The problem of approximately counting independent sets in graphs has been widely studied, see for example =-=[9, 12, 22, 23, 27]-=-, but the only previous work on the approximate counting of independent sets in hypergraphs seems to that of Dyer and Greenhill [12]. They showed rapid mixing to the uniform distribution of a simple M... |

62 | On counting independent sets in sparse graphs
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Citation Context ...ill usually give a tighter bound on mixing time than [17, Corollary 3]. See Remark 2.3 below. The problem of approximately counting independent sets in graphs has been widely studied, see for example =-=[9, 12, 22, 23, 27]-=-, but the only previous work on the approximate counting of independent sets in hypergraphs seems to that of Dyer and Greenhill [12]. They showed rapid mixing to the uniform distribution of a simple M... |

58 | Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem
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Citation Context .... Counting q-colourings of hypergraphs was considered by Bubley [2], who showed that the Glauber dynamics was rapidly mixing if q >= 2\Delta , generalising a result of Jerrum [19] and Salas and Sokal =-=[25]-=- for graphs. Much work has been done on improving this result for graph colourings, see [8] and its references, but little attention appears to have been given to the hypergraph case. Here we prove ra... |

51 | A new multilayered PCP and the hardness of hypergraph vertex cover
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Citation Context ...rings in hypergraphs with upperbounded degree, and lower-bounded edge size. These problems in hypergraphs are of interest in their own right but, while approximate optimisation has received attention =-=[6, 7, 18, 21]-=-, there has been surprisingly little work on approximate counting. Our results are achieved by considering, in the path coupling setting, the stopping time at which the distance between two coupled ch... |

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Citation Context ...ill usually give a tighter bound on mixing time than [17, Corollary 3]. See Remark 2.3 below. The problem of approximately counting independent sets in graphs has been widely studied, see for example =-=[9, 12, 22, 23, 27]-=-, but the only previous work on the approximate counting of independent sets in hypergraphs seems to that of Dyer and Greenhill [12]. They showed rapid mixing to the uniform distribution of a simple M... |

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Citation Context ...versa. For a path coupling P with respect to S ⊆ Ω 2 , we define β(M, P) = max (x,y)∈S EP[d(Xt+1, Yt+1)/d(x, y) | Xt = x, Yt = y]. The main path coupling method is then given by the following theorem =-=[11]-=-. Theorem 1.1. Let d be an integer valued metric defined on Ω×Ω which takes values in {0, . . . , D}. Let S be a subset of Ω×Ω such that for all (Xt, Yt) ∈ Ω×Ω there exists a path Xt = Z0, Z1, . . . ,... |

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Citation Context ...dge e 2 E, there are 2m-2 - 1 independent assignments to ue1, . . . , uem-2 if v1, v2 2 I and 2m-2 otherwise. This is equivalent to evaluating the partition function of a weighted H-colouring problem =-=[5, 13]-=- on G, with weight matrix A = ^2 m-2 2m-2 2m-2 2m-2 - 1* . The #P-completeness of H-colouring with this weight matrix follows either directly from [5] or indirectly from [13, Corollary 3.2]. The degre... |

32 |
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Citation Context ...dge e 2 E, there are 2m-2 - 1 independent assignments to ue1, . . . , uem-2 if v1, v2 2 I and 2m-2 otherwise. This is equivalent to evaluating the partition function of a weighted H-colouring problem =-=[5, 13]-=- on G, with weight matrix A = ^2 m-2 2m-2 2m-2 2m-2 - 1* . The #P-completeness of H-colouring with this weight matrix follows either directly from [5] or indirectly from [13, Corollary 3.2]. The degre... |

28 |
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Citation Context ...ion 1.1, independent sets in a hypergraph with \Deltas= 2 correspond to edge covers in a graph. Counting these is #P-complete, even for graphs with arbitrarily large minimum degree. This is stated in =-=[3]-=- but without proof, so we provide a proof in Appendix A. We now consider \Deltas>= 3. (The case m = \Deltas= 3 is discussed in [12].) Take a graph G = (V, E), and construct a hypergraph G = (V, E) by ... |

25 | A note on the Glauber dynamics for sampling independent sets, Electron
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Citation Context |

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Citation Context ...uber dynamics was rapidly mixing if q >= 2\Delta , generalising a result of Jerrum [19] and Salas and Sokal [25] for graphs. Much work has been done on improving this result for graph colourings, see =-=[8]-=- and its references, but little attention appears to have been given to the hypergraph case. Here we prove rapid mixing of Glauber dynamics for proper colourings of hypergraphs if m >= 4, q > \Delta ,... |

21 | An extension of path coupling and its application to the Glauber dynamics for graph colorings
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(Show Context)
Citation Context ...en the two chains has decreased at this stopping time, then the chain mixes rapidly. The first application of stopping times to path coupling was by Dyer, Goldberg, Greenhill, Jerrum and Mitzenmacher =-=[10]-=-. Their analysis was later improved by Hayes and Vigoda [17], using a method closely related to that developed this paper. Here we give a further improvement, yet with a simpler proof than that of [17... |

20 | The hardness of 3-Uniform hypergraph coloring
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(Show Context)
Citation Context ...rings in hypergraphs with upperbounded degree, and lower-bounded edge size. These problems in hypergraphs are of interest in their own right but, while approximate optimisation has received attention =-=[6, 7, 18, 21]-=-, there has been surprisingly little work on approximate counting. Our results are achieved by considering, in the path coupling setting, the stopping time at which the distance between two coupled ch... |

17 | CandJerrum M, On approximately counting colourings of small degree graphs - Bubley, Dyer, et al. - 1999 |

17 | Approximating coloring and maximum independent sets in 3-uniform hypergraphs
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Citation Context ...rings in hypergraphs with upperbounded degree, and lower-bounded edge size. These problems in hypergraphs are of interest in their own right but, while approximate optimisation has received attention =-=[6, 7, 18, 21]-=-, there has been surprisingly little work on approximate counting. Our results are achieved by considering, in the path coupling setting, the stopping time at which the distance between two coupled ch... |

15 |
Improved Approximation Lower Bounds on Small Occurrence Optimization Problems, ECCC
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Citation Context ...ts of size i (i = 0, 2, . . ., n). For * > 0 let ZG(*) = Pni=0 Ni*i define the hard core partition function. The following is a combination of results in Luby and Vigoda [22] and Berman and Karpinski =-=[1]-=-. Theorem 4.2. If * > 694/\Delta , there is no fpras for ZG(*) unless NP =RP. Proof. Let " be a constant such that the size of the largest independent set in a graph of maximum degree 4 cannot be appr... |

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Citation Context ...wing reduction from graph colouring. Let G = (V, E) be a graph with degree \Delta G, and 2 < q <= 3\Delta G/4. Without loss, we may assume \Delta G = d4q/3e. Colouring G with q colours is NP-complete =-=[14]-=-. For each edge e = {v1, v2} 2 E, let Sei = {uei1, uei2, . . . , ueim} (i = 1, 2, . . ., q) and Ve0 = Sqi=1 Sei . Let Ee0 comprise all subsets of V e0 of size m other than Sei (i = 1, 2, . . ., q). We... |

15 | Approximating maximum independent sets in uniform hypergraphs
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(Show Context)
Citation Context |

12 |
Randomized Algorithms: Approximation, Generation, and Counting
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Citation Context ...nt. We have no strong belief that either is close to the threshold at which approximate counting is possible, if such a threshold exists. Counting q-colourings of hypergraphs was considered by Bubley =-=[2]-=-, who showed that the Glauber dynamics was rapidly mixing if q >= 2\Delta , generalising a result of Jerrum [19] and Salas and Sokal [25] for graphs. Much work has been done on improving this result f... |

12 | Colouring graphs when the number of colours is nearly the maximum degree
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- 2001
(Show Context)
Citation Context ...G(m, \Delta ) has any q-colouring is NP-complete for any m > 1 and 2 < q <= (1 - 1/m)\Delta 1/(m-1). Proof. If m = 2, this is graph colouring, and the result follows from [14, Theorem 1.4]. (See also =-=[24]-=-.) For m >= 3, we use the following reduction from graph colouring. Let G = (V, E) be a graph with degree \Delta G, and 2 < q <= 3\Delta G/4. Without loss, we may assume \Delta G = d4q/3e. Colouring G... |

9 |
Very rapidly mixing Markov Chains for 2∆-colourings and for independent sets in a 4-regular graph, 2001 Random Struct
- Molloy
(Show Context)
Citation Context |

5 |
Variable length path coupling
- Hayes, Vigoda
(Show Context)
Citation Context ...the chain mixes rapidly. The first application of stopping times to path coupling was by Dyer, Goldberg, Greenhill, Jerrum and Mitzenmacher [10]. Their analysis was later improved by Hayes and Vigoda =-=[17]-=-, using a method closely related to that developed this paper. Here we give a further improvement, yet with a simpler proof than that of [17]. Technically, our method does not assume a bounded stoppin... |

2 |
Path coupling as a branching process, unpublished manuscript
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- 2002
(Show Context)
Citation Context ...Goldberg, Greenhill, Jerrum and Mitzenmacher [10]. Their analysis was later improved by Hayes and Vigoda [17], using a method closely related to that developed in this paper. Mitzenmacher and Niklova =-=[24]-=- had earlier stated a similar theorem, but could not provide a conclusive proof. Their approach had many simliarities to our Theorem 2.1, but conditioning problems necessitate a proof along somewhat d... |

1 |
16] C. Greenhill, The complexity of counting colourings and independent sets in sparse graphs and hypergraphs, Computational Complexity 9
- Garey, Johnson, et al.
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(Show Context)
Citation Context ...1n . Thus we obtain rapid mixing only in the case \Deltas= 2. This case has some intrinsic interest, since the complement of an independent set corresponds, under hypergraph duality, to an edge cover =-=[15]-=- in a graph. Thus we may uniformly generate edge covers, but the scope for unmodified path coupling is obviously severely limited. The insight on which this paper is based is as follows. Although in o... |

1 | Very rapidly mixing Markov chains for 2\Delta -coloring and for independent sets in a graph with maximum degree 4, Random Structures and Algorithms 18
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(Show Context)
Citation Context |

1 | Sequence A006129, The on-line encyclopedia of integer sequences - Sloane - 2004 |

1 |
On Markov chains for independent sets, Journal of Algorithms 35
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Citation Context ... of Computer Science, University of Bonn, 53117 Bonn, Germany. Email: marek@cs.uni-bonn.de. 1The problem of approximately counting independent sets in graphs has been widely studied, see for example =-=[9, 12, 23, 25, 29]-=-, but the only previous work on the approximate counting of independent sets in hypergraphs seems to be that of Dyer and Greenhill [12]. They showed rapid mixing to the uniform distribution of a simpl... |