We know that #SAT is #P-complete. We also know that problems with polynomial decision versions like PERMANENT are #P-complete. Is it true that finding the number of simple cycles in a graph, i.e. #CYCLE is also #P-complete? This is also a problem with decision version solved in polynomial time.

I suppose that it is either not-known or false, because it might be used a a popular example in textbooks like Arora and Barak's book, etc.


1 Answer 1


This is one of the problems (very briefly) discussed by Valiant, "The Complexity of Enumeration and Reliability Problems", SIAM J. Comput. 1979, doi:10.1137/0208032. See the mention of "elementary cycles" near the top of p. 417. Valiant gives a Turing reduction from the number of Hamiltonian cycles.


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