Sharp-P

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.

An NP 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 formula? (Boolean satisfiability problem)

The corresponding #P 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?

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.

Surprisingly, some #P problems that are believed to be difficult correspond to easy P problems. For more information on this, see #P-complete.

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 instead asks for the least significant bit of the #P answer.

The complexity class #P was first defined by Leslie Valiant in a 1979 article on the computation of the permanent, in which he proved that permanent is #P-complete.^[1]

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 $(1-\epsilon) P(a) \leq x \leq (1+\epsilon) P(a)$.^[2] The runtime of the algorithm is polynomial in a and 1/ε. The algorithm is based on leftover hash lemma.

References

1. 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.
2. Larry Stockmeyer (November 1985), "On Approximation Algorithms for #P" (PDF), SIAM J. Comput., 14 (4), lay summary

External links

• Complexity Zoo: Class #P