Improved mixing for the convex polygon triangulation flip walk
Abstract
We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n^3 log^3 n), the first progress since McShine and Tetali's O(n^5 log n) bound in 1997. In the process we give lower and upper bounds of respectively Omega(1/(sqrt n log n)) and O(1/sqrt n) -- asymptotically tight up to an O(log n) factor -- for the expansion of the associahedron graph K_n. The upper bound recovers Molloy, Reed, and Steiger's Omega(n^{3/2}) bound on the mixing time of the walk. To obtain these results, we introduce a framework consisting of a set of sufficient conditions under which a given Markov chain mixes rapidly. This framework is a purely combinatorial analogue that in some circumstances gives better results than the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda. In particular, in addition to the result for triangulations, we show quasipolynomial mixing for the k-angulation flip walk on a convex point set, for fixed k >= 4.
Cite
@article{arxiv.2207.09972,
title = {Improved mixing for the convex polygon triangulation flip walk},
author = {David Eppstein and Daniel Frishberg},
journal= {arXiv preprint arXiv:2207.09972},
year = {2023}
}