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 \({n \choose k/2}\) and \({n \choose mk/2}\), up to a factor polynomial in n, k, and m; in polynomial space, the bounds hold if multiplied by 3k/2 or 5mk/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(n ℓ) basic operations, where ℓ is the number of members in the two families and their subsets.
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Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M. (2009). Counting Paths and Packings in Halves. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_52
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DOI: https://doi.org/10.1007/978-3-642-04128-0_52
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