## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 17 (2012), paper no. 5, 6 pp.

### A counterexample to rapid mixing of the Ge-Štefankovič process

Leslie Goldberg and Mark Jerrum

#### Abstract

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.

#### Article information

**Source**

Electron. Commun. Probab., Volume 17 (2012), paper no. 5, 6 pp.

**Dates**

Accepted: 16 January 2012

First available in Project Euclid: 7 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1465263138

**Digital Object Identifier**

doi:10.1214/ECP.v17-1712

**Mathematical Reviews number (MathSciNet)**

MR2872574

**Zentralblatt MATH identifier**

1246.60094

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Secondary: 05C31 05C69 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15]

**Keywords**

Glauber dynamics Independent sets in graphs Markov chains Mixing time Randomised algorithms

**Rights**

This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Goldberg, Leslie; Jerrum, Mark. A counterexample to rapid mixing of the Ge-Štefankovič process. Electron. Commun. Probab. 17 (2012), paper no. 5, 6 pp. doi:10.1214/ECP.v17-1712. https://projecteuclid.org/euclid.ecp/1465263138