The fast intersection transform with applications to counting paths

We present an algorithm for evaluating a linear “intersection transform” of a function defined on the lattice of subsets of an n-element set. In particular, the algorithm constructs an arithmetic circuit for evaluating the transform in “down-closure time” relative to the support of the function and the evaluation domain. As an application, we develop an algorithm that, given as input a digraph with n vertices and bounded integer weights at the edges, counts paths by weight and given length 0 ≤ ` ≤ n − 1 in time O∗(exp(n ·H(`/(2n)))), where H(p) = −p log p− (1− p) log(1− p), and the notation O∗(·) suppresses a factor polynomial in n.