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I have been told that there are some good polynomial time algorithms for approximating the number of simple paths in an directed graph from given starting vertex $s$ to given ending vertex $t$. Does anyone know of a good reference on this subject?

Background: counting the exact number of paths in a general graph is #P-complete but there may exist polynomial time approximations for the problem. I'm especially interested in random approximations.

Thanks in advance.

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I had the same problem e solve using Random Walk. –  user9913 Jun 28 '12 at 14:44
@bbejot: See How hard is counting the number of simple paths between two nodes in a directed graph? The only answer, by jmad, provides a link to a paper that provides indeed a random approximation –  Carlos Linares López Nov 26 '12 at 22:11
Not sure if this helps, but you could use Balanced Families of Perfect Hash to approximate the number of simple paths of length $k$ in $2^{O(klogk)}$ which might be useful if you're looking for counting short paths. –  R B Jan 19 '14 at 16:47

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