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 ${ n \choose k/2}$ and ${ n \choose mk/2}$, 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.
Categories: cs.DS cs.DM