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In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. [1] Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines.
If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. [21] Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai, runs in quasi-polynomial ...
The class P #P consists of all the problems that can be solved in polynomial time if you have access to instantaneous answers to any counting problem in #P (polynomial time relative to a #P oracle). Thus Toda's theorem implies that for any problem in the polynomial hierarchy there is a deterministic polynomial-time Turing reduction to a ...
The union of the classes in the polynomial hierarchy: P NP: Solvable in polynomial time with an oracle for a problem in NP; also known as Δ 2 P PP: Probabilistically Polynomial (answer is right with probability slightly more than 1/2) PPAD: Polynomial Parity Arguments on Directed graphs PR: Solvable by recursively building up arithmetic ...
Pictorial representation of the polynomial time hierarchy. The arrows denote inclusion. In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms. It involves the research ...
In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: = PH was first defined by Larry Stockmeyer. [1] It is a special case of hierarchy of bounded alternating Turing machine.
For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√6 are algebraic numbers.
However, unless P=NP, any polynomial-time algorithm must asymptotically be wrong on more than polynomially many of the exponentially many inputs of a certain size. [14] "If P=NP, all cryptographic ciphers can be broken." A polynomial-time problem can be very difficult to solve in practice if the polynomial's degree or constants are large enough.