Ge, Qi ;
Stefankovic, Daniel
A graph polynomial for independent sets of bipartite graphs
Abstract
We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).
We analyze the complexity of exact evaluation of the polynomial at rational points and show a dichotomy resultfor most points exact evaluation is #Phard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.
We propose a natural Markov chain to approximately evaluate the polynomial for a range of parameters. We prove an upper bound on the mixing time of the Markov chain on trees. As a byproduct we show that the ``single bond flip'' Markov chain for the random cluster model is rapidly mixing on constant treewidth graphs.
BibTeX  Entry
@InProceedings{ge_et_al:LIPIcs:2010:2867,
author = {Qi Ge and Daniel Stefankovic},
title = {{A graph polynomial for independent sets of bipartite graphs}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)},
pages = {240250},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897231},
ISSN = {18688969},
year = {2010},
volume = {8},
editor = {Kamal Lodaya and Meena Mahajan},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2867},
URN = {urn:nbn:de:0030drops28676},
doi = {http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2010.240},
annote = {Keywords: graph polynomials, #Pcomplete, independent sets, approximate counting problems, Markov chain Monte Carlo}
}
2010
Keywords: 

graph polynomials, #Pcomplete, independent sets, approximate counting problems, Markov chain Monte Carlo 
Seminar: 

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

Related Scholarly Article: 


Issue date: 

2010 
Date of publication: 

2010 