Metric Construction, Stopping Times and Path Coupling

In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to stopping time analysis. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling with stopping times, using a metric which allows us to restrict analysis to standard one-step path coupling. This approach provides insight for the design of nonstandard metrics giving improvements in the analysis of specific problems. We give illustrative applications to hypergraph independent sets and SAT instances, hypergraph colourings and colourings of bipartite graphs. In particular we prove rapid mixing for Glauber dynamics on independent sets in hypergraphs whenever the minimum edge size m and degree ∆ satisfy m ≥ ∆ + 2, and for all edge sizes when ∆ = 3. Previously rapid mixing was only known for m ≥ 2∆ + 1. This result leads to approximation schemes for monotone SAT formulae in which the maximum number of occurrences of a variable (∆) and the minimum number of variables per clause (m) satisfy the same condition. For Glauber dynamics on proper colourings of 3-uniform hypergraphs we prove rapid mixing whenever the number of colours q is at least ⌈ 3 2 ∆+ 1 ⌉ . Previously the best known result was for q ≥ 1.65∆ and ∆ ≥ ∆0 for some large ∆0. Finally we prove rapid mixing of scan dynamics (where the order of vertex updates is deterministic) for proper colourings of bipartite graphs whenever q > f(∆), where f(∆) → β∆, as ∆ → ∞, and β satisfies 1 β e 1/β = 1, (β ≈ 1.76). This gives rapid mixing with fewer colours than Vigoda’s 11∆/6 bound [22], whenever ∆ ≥ 31, and equals this bound for ∆ ≥ 14. School of Computing, University of Leeds, Leeds LS2 9JT, UK. Email: {dyer,magnusb}@comp.leeds.ac.uk. Dept. of Computer Science, University of Bonn, 53117 Bonn, Germany. Email: [email protected]. 1