Balanced hashing, color coding and approximate counting

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 (ε,k)-balanced family of hash functions, if there exists a positive T so that for every K ⊂ [n] of size |K| = k, the number of functions in the family that are one-to-one on K is between (1 - ε)T and (1 + ε)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 ^[k/2], and that for every ε > 1/poly(k) there are explicit constructions of (ε,k)-balanced families of hash functions from [n] to [k] of size e^(1+o(1))k logn. 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 ε, for all k ≤ O(logn).