Complexity of counting subgraphs: Only the boundedness of the vertexcover number counts
Curticapean, Radu and Marx, Dániel (2014) Complexity of counting subgraphs: Only the boundedness of the vertexcover number counts. In: 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014. IEEE Computer Society, Los Alamitos, pp. 130139. ISBN 9781479965175 10.1109/FOCS.2014.22

Text
Curticapean_130_2835236_ny.pdf Download (814kB)  Preview 
Abstract
For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertexcover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomialtime solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertexcover number, then #Sub(C) is #W[1]hard parameterized by the size of H and hence not polynomialtime solvable and not even fixedparameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting kmatchings in bipartite graphs is #W[1]hard. Recently, Curticapean [ICALP 2013] proved the #W[1]hardness of counting kmatchings in general graphs, our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f(k)∗no(k/log(k)) time algorithm for counting kmatchings in bipartite graphs for any computable function f. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]hard, as well as a similar almosttight ETHbased lower bound on the exponent.
Item Type:  Book Section 

Subjects:  Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány 
Divisions:  Informatics Laboratory 
SWORD Depositor:  MTMT Injector 
Depositing User:  MTMT Injector 
Date Deposited:  12 Feb 2015 19:17 
Last Modified:  16 Feb 2016 19:00 
URI:  http://eprints.sztaki.hu/id/eprint/8203 
Update Item 