Switch-based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

TL;DR

This paper proves that switch-based Markov chains can efficiently sample Hamiltonian cycles in dense graphs with high minimum degree, demonstrating irreducibility and rapid mixing in certain graph classes.

Contribution

It establishes the irreducibility of switch Markov chains for dense graphs with minimum degree above n/2+7 and shows rapid mixing on dense monotone graphs.

Findings

Switch operations of size at most 10 connect all Hamiltonian cycles in dense graphs.

The switch Markov chain is irreducible for graphs with minimum degree ≥ n/2+7.

Rapid mixing is demonstrated on dense monotone graphs.

Abstract

We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most $10$, implying that the switch Markov chain using switches of size at most $10$ is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.