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DOI: 10.4230/LIPIcs.FSTTCS.2010.240
URN: urn:nbn:de:0030-drops-28676
URL: https://drops.dagstuhl.de/opus/volltexte/2010/2867/
Ge, Qi ; Stefankovic, Daniel
A graph polynomial for independent sets of bipartite graphs
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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 result---for most points exact evaluation is #P-hard (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 by-product we show that the ``single bond flip'' Markov chain for the random cluster model is rapidly mixing on constant tree-width 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 = {240--250},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-23-1},
ISSN = {1868-8969},
year = {2010},
volume = {8},
editor = {Kamal Lodaya and Meena Mahajan},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2867},
URN = {urn:nbn:de:0030-drops-28676},
doi = {10.4230/LIPIcs.FSTTCS.2010.240},
annote = {Keywords: graph polynomials, #P-complete, independent sets, approximate counting problems, Markov chain Monte Carlo}
}
| Keywords: | graph polynomials, #P-complete, independent sets, approximate counting problems, Markov chain Monte Carlo | |
| Seminar: | IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010) | |
| Issue date: | 2010 | |
| Date of publication: | 14.12.2010 |
