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.
@InProceedings{eppstein_et_al:LIPIcs.ICALP.2023.56, author = {Eppstein, David and Frishberg, Daniel}, title = {{Improved Mixing for the Convex Polygon Triangulation Flip Walk}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {56:1--56:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.56}, URN = {urn:nbn:de:0030-drops-181081}, doi = {10.4230/LIPIcs.ICALP.2023.56}, annote = {Keywords: associahedron, mixing time, mcmc, Markov chains, triangulations, quadrangulations, k-angulations, multicommodity flow, projection-restriction} }
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