Sampling Eulerian orientations of triangular lattice graphs

Creed, Páidí

doi:10.1016/j.jda.2008.09.006

Cited by 7 publications

(6 citation statements)

References 27 publications

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“…Not surprisingly, counting α-orientations is #P -complete. Namely, consider an undirected Eulerian graph G (with all even degrees); the α-orientations of G, where α(v) = d(v)/2, correspond precisely to Eulerian orientations of G. The latter problem has been shown to be #P -complete by Mihail and Winkler [22], and more recently Creed [8] showed that it remains #P -complete even when restricted to the class of planar graphs.…”

Section: -Orientationsmentioning

confidence: 99%

Sampling and Counting 3-Orientations of Planar Triangulations

Miracle¹,

Randall²,

Streib³

et al. 2016

SIAM J. Discrete Math.

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Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing" chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that, when restricted to planar triangulations of maximum degree six, this Markov chain is rapidly mixing and we can approximately count 3-orientations. Next, we construct a triangulation with high degree on which this Markov chain mixes slowly. Finally, we consider an "edge-flipping" chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. We prove that this chain is always rapidly mixing.1. Introduction. The 3-orientations of a graph have given rise to beautiful combinatorics and computational applications. A 3-orientation of a planar triangulation is an orientation of the internal edges of the triangulation such that every internal vertex has out-degree three. We study natural Markov chains for sampling 3-orientations in two contexts, when the triangulation is fixed and when we consider the union of all planar triangulations on a fixed number of vertices. When the triangulation is fixed, we allow moves that reverse the orientation of edges around a triangle if they form a directed cycle. We show that the chain is rapidly mixing (converging in polynomial time to equilibrium) if the maximum degree of the triangulation is six, but can be slowly mixing (requiring exponential time) if the degrees are unbounded. When the maximum degree of the triangulation is six, we give a FPRAS (fully polynomial randomized approximation scheme) for approximately counting the number of 3-orientations of the fixed triangulation. To sample from the set of all 3-orientations of triangulations with n vertices we use a simple "edge-flipping" chain and show it is always rapidly mixing. These chains arise in contexts such as sampling Eulerian orientations and triangulations of fixed planar point sets, so there is additional motivation for understanding their convergence rates.

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Section: -Orientationsmentioning

confidence: 99%

Sampling and Counting 3-Orientations of Planar Triangulations

Miracle¹,

Randall²,

Streib³

et al. 2016

SIAM J. Discrete Math.

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“…The results of this paper have recently been used to show rapid mixing of a Markov chain on the set of Eulerian orientations of a triangular subsection G of the triangular lattice [4]. The Markov chain M o used is as follows.…”

Section: Application: Sampling Eulerian Orientations Of the Triangulamentioning

confidence: 99%

“…The additional moves allowed in M o involve reversing the edges bounding several adjacent faces with some probability, in the event that the face selected at random does not form a directed cycle. In [4] it is shown that β( M o ) 1. An appeal to Theorem 2 of this paper completes the proof of rapid mixing, avoiding an involved direct proof bounding σ 2 ( M o ).…”

Section: Application: Sampling Eulerian Orientations Of the Triangulamentioning

confidence: 99%

“…An appeal to Theorem 2 of this paper completes the proof of rapid mixing, avoiding an involved direct proof bounding σ 2 ( M o ). See [4] for full details.…”

Section: Application: Sampling Eulerian Orientations Of the Triangulamentioning

confidence: 99%

“…Where this theorem does not apply, we give a modification of this idea which will often be applicable. We illustrate our methods with examples taken from the literature [2,4,10,15] which have made use of the non-expansion case of path coupling. In all these examples we improve the bound on mixing time established in the source paper.…”

Section: Introductionmentioning

confidence: 99%

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Path coupling without contraction

Bordewich

Dyer

2007

Journal of Discrete Algorithms

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Path coupling is a useful technique for simplifying the analysis of a coupling of a Markov chain. Rather than defining and analysing the coupling on every pair in Ω × Ω, where Ω is the state space of the Markov chain, analysis is done on a smaller set S ⊆ Ω × Ω. If the coefficient of contraction β is strictly less than one, no further analysis is needed in order to show rapid mixing. However, if β = 1 then analysis (of the variance) is still required for all pairs in Ω × Ω. In this paper we present a new approach which shows rapid mixing in the case β = 1 with a further condition which only needs to be checked for pairs in S, greatly simplifying the work involved. We also present a technique applicable when β = 1 and our condition is not met.

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