Sampling and Counting 3-Orientations of Planar Triangulations

Abstract

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.

Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

“…In [MRST16] and in this paper there are proofs for rapid mixing of the face flip Markov chain for αorientations on graphs with small constant maximum degree and negative results in the sense of slow mixing of these Markov chains for graphs with large maximum degree. Could it be that the face flip Markov chain for α-orientations is rapidly mixing for all graphs of small maximum degree?…”

“…The basic approach for our analysis of M 2T on low degree quadrangulations is similar to what Fehrenbach and Rüschendorf [FR04] did on a class of subgraphs of the quadrangular grid. In the context of 3-orientations of triangulations similar methods were applied by Creed [Cre09] to certain subgraphs of the triangular grid and later by Miracle et al [MRST16] to general triangulations. As Creed [Cre09] noted there is an inaccurate claim in the proof of [FR04].…”

“…As Creed [Cre09] noted there is an inaccurate claim in the proof of [FR04]. Later Miracle et al [MRST12] stepped into the same trap (it has been corrected in the final version [MRST16]). In 3.2.1 below we discuss these issues and show how to repair the proofs.…”

“…From the correspondence between Schnyder woods and 3-orientations it follows that the triangle flip Markov chain can be used to sample from either of these structures. The mixing time of this Markov chain was studied by Creed [Cre09] for certain subgraphs of the triangular grid and then By Miracle et al [MRST16] for general triangulations. Here we want to revisit the following negative result.…”

Morphing Schnyder Drawings of Planar Triangulations

“…We have attributed the following result to Brehm [7], but he does not state it as a single result. It helps to read Miracle et al [17] and Eppstein et al [10].…”