Counting paths and packings in halves

Abstract

We show 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 and , 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 O(nℓ) basic operations, where ℓ is the number of members in the two families and their subsets. © 2009 Springer Berlin Heidelberg.