On systematic scan for sampling H-colourings of the path

TL;DR

This paper demonstrates that systematic scan Markov chains efficiently sample H-colourings of a path in logarithmic time, significantly improving previous bounds, and also provides results for random update chains.

Contribution

It proves that systematic scan Markov chains mix in O(log n) scans for H-colourings of paths, a substantial improvement over prior O(n^5) bounds, and extends results to specific H families.

Findings

Systematic scan mixes in O(log n) scans for any fixed H.

For certain H, systematic scan also mixes in O(log n) scans.

Random update chains mix in O(n log n) updates, improving previous bounds.

Abstract

This paper is concerned with sampling from the uniform distribution on H-colourings of the n-vertex path using systematic scan Markov chains. An H-colouring of the n-vertex path is a homomorphism from the n-vertex path to some fixed graph H. We show that systematic scan for H-colourings of the n-vertex path mixes in O(log n) scans for any fixed H. This is a significant improvement over the previous bound on the mixing time which was O(n^5) scans. Furthermore we show that for a slightly more restricted family of H (where any two vertices are connected by a 2-edge path) systematic scan also mixes in O(log n) scans for any scan order. Finally, for completeness, we show that a random update Markov chain mixes in O(n log n) updates for any fixed H, improving the previous bound on the mixing time from O(n^5) updates.