eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-07-05
56:1
56:17
10.4230/LIPIcs.ICALP.2023.56
article
Improved Mixing for the Convex Polygon Triangulation Flip Walk
Eppstein, David
1
Frishberg, Daniel
1
https://orcid.org/0000-0002-1861-5439
Department of Computer Science, University of California, Irvine, CA, USA
We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n³ log³ n), the first progress since McShine and Tetali’s O(n⁵ log n) bound in 1997. In the process we give lower and upper bounds of respectively Ω(1/(√n log n)) and O(1/√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 Ω(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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol261-icalp2023/LIPIcs.ICALP.2023.56/LIPIcs.ICALP.2023.56.pdf
associahedron
mixing time
mcmc
Markov chains
triangulations
quadrangulations
k-angulations
multicommodity flow
projection-restriction