Counting Paths and Packings in Halves

Abstract

It is shown that one can count $k$-edge paths in an $n$-vertex graph and $m$-set $k$-packings on an $n$-element universe, respectively, in time $\left(\genfrac{}{}{0ex}{}{n}{k/2}\right)$ and $\left(\genfrac{}{}{0ex}{}{n}{mk/2}\right)$, up to a factor polynomial in $n$, $k$, and $m$; in polynomial space, the bounds hold if multiplied by $3^{k/2}$ or $5^{mk/2}$, respectively. These are implications of a more general result: given two set families on an $n$-element universe, one can count the disjoint pairs of sets in the Cartesian product of the two families with $\nO(n \ell)$ basic operations, where $\ell$ is the number of members in the two families and their subsets.

References in corpus (1)

• The fast intersection transform with applications to counting paths