In computational complexity theory, the complexity class #P (pronounced "sharp P" or, sometimes "number P" or "hash P") is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute f(x)", where f is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems.
Relation to decision problems
An NP decision problem is often of the form "Are there any solutions that satisfy certain constraints?" For example:
- Are there any subsets of a list of integers that add up to zero? (subset sum problem)
- Are there any Hamiltonian cycles in a given graph with cost less than 100? (traveling salesman problem)
- Are there any variable assignments that satisfy a given CNF (conjunctive normal form) formula? (Boolean satisfiability problem or SAT)
The corresponding #P function problems ask "how many" rather than "are there any". For example:
- How many subsets of a list of integers add up to zero?
- How many Hamiltonian cycles in a given graph have cost less than 100?
- How many variable assignments satisfy a given CNF formula?
How hard is that?
Clearly, a #P problem must be at least as hard as the corresponding NP problem. If it's easy to count answers, then it must be easy to tell whether there are any answers—just count them and see whether the count is greater than zero.
One consequence of Toda's theorem is that a polynomial-time machine with a #P oracle (P#P) can solve all problems in PH, the entire polynomial hierarchy. In fact, the polynomial-time machine only needs to make one #P query to solve any problem in PH. This is an indication of the extreme difficulty of solving #P-complete problems exactly.
The closest decision problem class to #P is PP, which asks whether a majority (more than half) of the computation paths accept. This finds the most significant bit in the #P problem answer. The decision problem class ⊕P (pronounced "Parity-P") instead asks for the least significant bit of the #P answer.
Larry Stockmeyer has proved that for every #P problem P there exists a randomized algorithm using an oracle for SAT, which given an instance a of P and ε > 0 returns with high probability a number x such that . The runtime of the algorithm is polynomial in a and 1/ε. The algorithm is based on the leftover hash lemma.
- Leslie G. Valiant (1979). "The Complexity of Computing the Permanent". Theoretical Computer Science. Elsevier. 8 (2): 189–201. doi:10.1016/0304-3975(79)90044-6.
- Stockmeyer, Larry (November 1985). "On Approximation Algorithms for #P" (PDF). SIAM Journal on Computing. 14 (4): 849. doi:10.1137/0214060. Archived from the original (PDF) on 2009. Lay summary.