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 \emph{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 non-standard 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. 
Magnus Bordewich, Martin Dyer, Marek Karpinski