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.
@InProceedings{ge_et_al:LIPIcs.FSTTCS.2010.240, author = {Ge, Qi and Stefankovic, Daniel}, 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 = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.240}, 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} }
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