Open Access
Translator Disclaimer
2012 A counterexample to rapid mixing of the Ge-Štefankovič process
Leslie Goldberg, Mark Jerrum
Author Affiliations +
Electron. Commun. Probab. 17: 1-6 (2012). DOI: 10.1214/ECP.v17-1712


Ge and Štefankovič have recently introduced a Markov chain which, if rapidly mixing, would provide an efficientprocedure for sampling independent sets in a bipartite graph. Such a procedure would be a breakthrough because it would give an efficient randomised algorithm for approximately counting independent sets in a bipartite graph, which would in turn imply the existence of efficient approximation algorithms for a number of significant counting problems whose computational complexity is so far unresolved. Their Markov chain is based on a novel two-variable graph polynomial which, when specialised to a bipartite graph, and evaluated at the point (1/2,1), givesthe number of independent sets in the graph. The Markov chain is promising, in the sense that it overcomes the most obvious barrier to rapid mixing. However, we show here, by exhibiting a sequence of counterexamples, that its mixing timeis exponential in the size of the input when the input is chosen from a particular infinite family of bipartite graphs.


Download Citation

Leslie Goldberg. Mark Jerrum. "A counterexample to rapid mixing of the Ge-Štefankovič process." Electron. Commun. Probab. 17 1 - 6, 2012.


Accepted: 16 January 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1246.60094
MathSciNet: MR2872574
Digital Object Identifier: 10.1214/ECP.v17-1712

Primary: 60J10
Secondary: 05C31 , 05C69 , 68Q17

Keywords: Glauber dynamics , Independent sets in graphs , Markov chains , mixing time , randomised algorithms


Back to Top