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 $0\leq\ell\leq n-1$ in time $O^(\exp(n\cdot H(\ell/(2n))))$, where $H(p)=-p\log p-(1-p)\log(1-p)$, and the notation $O^(\cdot)$ suppresses a factor polynomial in $n$.

💡 Research Summary

The paper introduces a novel linear operator on the Boolean lattice of subsets, called the intersection transform. Given two functions $f,g:2^{