Complexity of counting subgraphs: only the boundedness of the vertexcover number counts
Abstract
For a class $\mathcal{H}$ of graphs, #Sub$(\mathcal{H})$ is the counting problem that, given a graph $H\in \mathcal{H}$ and an arbitrary graph $G$, asks for the number of subgraphs of $G$ isomorphic to $H$. It is known that if $\mathcal{H}$ has bounded vertexcover number (equivalently, the size of the maximum matching in $\mathcal{H}$ is bounded), then #Sub$(\mathcal{H})$ is polynomialtime solvable. We complement this result with a corresponding lower bound: if $\mathcal{H}$ is any recursively enumerable class of graphs with unbounded vertexcover number, then #Sub$(\mathcal{H})$ is #W[1]hard parameterized by the size of $H$ and hence not polynomialtime solvable and not even fixedparameter tractable, unless FPT = #W[1]. As a first step of the proof, we show that counting $k$matchings in bipartite graphs is #W[1]hard. Recently, Curticapean [ICALP 2013] proved the #W[1]hardness of counting $k$matchings 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 $f(k)n^{o(k/\log k)}$ time algorithm for counting $k$matchings in bipartite graphs for any computable function $f(k)$. 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.2929
 Bibcode:
 2014arXiv1407.2929C
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 42 pages, 8 figures, 5 tables