16
$\begingroup$

Let $G$ be a digraph (not necessarily a DAG) and let $s,t \in V(G)$. What is the complexity of counting the number of simple $s-t$ paths in $G$.

I would expect the problem to be #${\mathsf P}$-complete but have not been able to locate an exact reference.

Also notice that a number of similar questions have been answered correctly here and elsewhere but not this precise question - to emphasise I am not interested in counting walks and/or undirected graphs (in the first case the variant is in ${\mathsf P}$ and in the other #${\mathsf P}$-hard).

$\endgroup$
  • $\begingroup$ The #P-completeness applies even for undirected graphs, and this was discussed before. Perhaps a more interesting question would be if this is known to be $APX$-hard. $\endgroup$ – R B Sep 8 '14 at 7:01
19
$\begingroup$

The #P-completeness proof of counting simple s-t paths in both undirected and directed graphs can be found in:

Leslie G. Valiant: The Complexity of Enumeration and Reliability Problems. SIAM J. Comput. 8(3): 410-421 (1979)

From the paper:

...
4. Some #P-complete problems
...
14. S-T PATHS (i.e. SELF AVOIDING WALKS) (directed or undirected)
Input: $G; s,t \in V$
Output: Number of (directed) paths from $s$ to $t$ that visit every node at most once.
...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.