TR09-012 Authors: Noga Alon, Shai Gutner

Publication: 6th February 2009 23:37

Downloads: 2973

Keywords:

Color Coding is an algorithmic technique for deciding efficiently

if a given input graph contains a path of a given length (or

another small subgraph of constant tree-width). Applications of the

method in computational biology motivate the study of similar

algorithms for counting the number of copies of a given subgraph.

While it is unlikely that exact counting of this type can be

performed efficiently, as the problem is $\#W[1]$-complete

even for paths,

approximate counting is possible, and leads to the investigation

of an intriguing variant of families of perfect hash functions. A

family of functions from $[n]$ to $[k]$ is an

$(\epsilon,k)$-balanced family of hash functions, if there exists

a positive $T$ so that for every $K \subset [n]$ of size $|K|=k$,

the number of functions in the family that are one-to-one on $K$

is between $(1-\epsilon)T$ and $(1+\epsilon)T$. The family is

perfectly $k$-balanced if it is $(0,k)$-balanced.

We show that every such perfectly $k$-balanced family is of size

at least $c(k) n^{\lfloor k/2 \rfloor}$, and that for every

$\epsilon>\frac{1}{poly(k)}$ there are explicit constructions of

$(\epsilon,k)$-balanced families of hash functions from $[n]$ to

$[k]$ of size $e^{(1+o(1))k} \log n$. This is tight up to the $o(1)$-term

in the exponent, and supplies deterministic

polynomial time algorithms for approximately counting the number

of paths or cycles of a specified length $k$ (or copies of any

graph $H$ with $k$ vertices and bounded tree-width) in a given

input graph of size $n$, up to relative error $\epsilon$, for all

$k \leq O(\log n)$.