Abstract
Jaeger, Vertigan, and Welsh (1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #Phard almost everywhere, and the remaining points admit polynomialtime algorithms. Dell, Husfeldt, and WahlĂ©n (2010) and Husfeldt and Taslaman (2010), in combination with the results of Curticapean (2015), extended the #Phardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y=1, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given nvertex graph cannot be determined in time exp(o(n)) unless #ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann (1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #Phard cases cannot be solved in time exp(o(n)) unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp(o(n)) unless #ETH fails.
In order to prove our results, we use the block interpolation idea by Curticapean (2015) and transfer it to systems of linear equations that might not directly correspond to interpolation.
BibTeX  Entry
@InProceedings{brand_et_al:LIPIcs:2017:6942,
author = {Cornelius Brand and Holger Dell and Marc Roth},
title = {{FineGrained Dichotomies for the Tutte Plane and Boolean #CSP}},
booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)},
pages = {9:19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770231},
ISSN = {18688969},
year = {2017},
volume = {63},
editor = {Jiong Guo and Danny Hermelin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/6942},
URN = {urn:nbn:de:0030drops69426},
doi = {10.4230/LIPIcs.IPEC.2016.9},
annote = {Keywords: computational complexity, counting problems, Tutte polynomial, exponential time hypothesis, forests, independent sets}
}
Keywords: 

computational complexity, counting problems, Tutte polynomial, exponential time hypothesis, forests, independent sets 
Collection: 

11th International Symposium on Parameterized and Exact Computation (IPEC 2016) 
Issue Date: 

2017 
Date of publication: 

09.02.2017 