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