In computational complexity theory, the complexity class #P (pronounced "number P" or, sometimes "sharp 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 ƒ(x)", where ƒ 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 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.
- Larry Stockmeyer (November 1985), "On Approximation Algorithms for #P" (PDF), SIAM J. Comput., 14 (4), lay summary