This problem is solved by the polynomial-time Horn-satisfiability algorithm, and is in fact P-complete.

Although this problem seems easier, it has been shown that if there is a practical ( randomized polynomial-time ) algorithm to solve this problem, then all problems in NP can be solved just as easily.

The AKS primality test, published in 2002, proves that primality testing also lies in P, while factorization may or may not have a polynomial-time algorithm.

Worse yet, since the aforementioned decision problem for CSG's is PSPACE-complete, that makes them totally unworkable for practical use, as a polynomial-time algorithm for a PSPACE-complete problem would imply P = NP.

In this sense the GCD problem is analogous to e. g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete.

) is NP-complete, thus it is expected that no algorithm can be both correct and fast ( polynomial-time ) on all cases.

* There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine, described below.

It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one half.

Jerrum, Valiant, and Vazirani showed that every # P-complete problem either has an FPRAS, or is essentially impossible to approximate ; if there is any polynomial-time algorithm which consistently produces an approximation of a # P-complete problem which is within a polynomial ratio in the size of the input of the exact answer, then that algorithm can be used to construct an FPRAS.

In the special case that distances between cities are all either one or two ( and thus the triangle inequality is necessarily satisfied ), there is a polynomial-time approximation algorithm that finds a tour of length at most 8 / 7 times the optimal tour length.

It is known that, unless P = NP, there is no polynomial-time algorithm that finds a tour of length at most 220 / 219 = 1. 00456 … times the optimal tour's length.

In general, for any c > 0, where d is the number of dimensions in the Euclidean space, there is a polynomial-time algorithm that finds a tour of length at most ( 1 + 1 / c ) times the optimal for geometric instances of TSP in time ; this is called a polynomial-time approximation scheme ( PTAS ).

In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer.

In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer.

This makes the calculation using the Legendre symbol significantly slower than the one using the Jacobi symbol, as there is no known polynomial-time algorithm for factoring integers .< ref > The number field sieve, the fastest known algorithm, requires operations to factor N. See Cohen, p. 495 </ ref > In fact, this is why Jacobi introduced the symbol.

That is, given the first < var > k </ var > bits of a random sequence, there is no polynomial-time algorithm that can predict the (< var > k </ var >+ 1 ) th bit with probability of success better than 50 %.

Using modular exponentiation by repeated squaring, the running time of this algorithm is O ( k log < sup > 3 </ sup > n ), where k is the number of different values of a we test ; thus this is an efficient, polynomial-time algorithm.

For general N, the Euclidean Steiner tree problem is NP-hard, and hence it is not known whether an optimal solution can be found by using a polynomial-time algorithm.

Therefore an algorithm that finds a minimum spanning tree is a polynomial-time factor-2 approximation algorithm for the metric Steiner tree problem.

Worse still, it is APX-complete, meaning there is no polynomial-time approximation scheme ( PTAS ) for this problem unless P = NP.

There are a large number of interesting tasks, and while it is not known if a polynomial-time solution exists for any of them, it is known that if such a solution exists for any of them, one exists for all of them.

Many people have tried to find classical polynomial-time algorithms for it and failed, and therefore it is widely suspected to be outside P.

The complexity class P is contained in NP, but NP contains many important problems, the hardest of which are called NP-complete problems, for which no polynomial-time algorithms are known for solving them ( although they can be verified in polynomial time ).

A language B is PSPACE-complete if it is in PSPACE and it is PSPACE-hard, which means for all A PSPACE, A B, where A B means that there is a polynomial-time many-one reduction from A to B. PSPACE-complete problems are of great importance to studying PSPACE problems because they represent the most difficult problems in PSPACE.

In his " Feasibly Constructive Proofs and the Propositional Calculus " paper published in 1975, he introduced the equational theory PV ( standing for Polynomial-time Verifiable ) to formalize the notion of proofs using only polynomial-time concepts.

Although it is widely suspected that there are no polynomial-time algorithms for NP-hard problems, this has never been proven.

The logic behind this is analogous to the logic that a polynomial-time solution to an NP-complete problem would prove P = NP: if we have a NC reduction from any problem in P to a problem A, and an NC solution for A, then NC = P. Similarly, if we have a log-space reduction from any problem in P to a problem A, and a log-space solution for A, then L = P.

The probablisitic polynomial-time Turing Machine V < sup >*</ sup >< sub > w </ sub >( x ) corresponds to the deterministic polynomial-time Turing Machine V ( x, w ) by replacing the random tape of V < sup >*</ sup > with a second input tape for V on which is written the sequence of coin flips.

By selecting the witness as a random string, the verifier is a probabilistic polynomial-time Turing Machine whose probability of accepting x when x is in X is large ( greater than 1 / 2, say ), but zero if x is not in X ( for RP ); of rejecting x when x is not in X is large but zero if x is in X ( for co-RP ); and of correctly accepting or rejecting x as a member of X is large, but zero of incorrectly accepting or rejecting x ( for ZPP ).

Some problems are known to be solvable in polynomial-time, but no concrete algorithm is known for solving them.

One consequence of Toda's theorem is that a polynomial-time machine with a # P oracle ( P < sup ># P </ sup >) 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.

A problem is # P-complete if and only if it is in # P, and every problem in # P can be reduced to it by a polynomial-time counting reduction, i. e. a polynomial-time Turing reduction relating the cardinalities of solution sets.

Many # P-complete problems have a fully polynomial-time randomized approximation scheme, or " FPRAS ," which, informally, will produce with high probability an approximation to an arbitrary degree of accuracy, in time that is polynomial with respect to both the size of the problem and the degree of accuracy required.

This is done by polynomial-time reduction from 3-SAT to the other problem.

A few years later, in his seminal 1971 paper " The Complexity of Theorem Proving Procedures ", Cook formalized the notions of polynomial-time reduction ( a. k. a. Cook reduction ) and NP-completeness, and proved the existence of an NP-complete problem by showing that the Boolean satisfiability problem ( usually known as SAT ) is NP-complete.

A decision problem C is co-NP-complete if it is in co-NP and if every problem in co-NP is polynomial-time many-one reducible to it.

* Since NP-complete problems transform to each other by polynomial-time many-one reduction ( also called polynomial transformation ), all NP-complete problems can be solved in polynomial time by a reduction to H, thus all problems in NP reduce to H ; note, however, that this involves combining two different transformations: from NP-complete decision problems to NP-complete problem L by polynomial transformation, and from L to H by polynomial Turing reduction ;

A language L is in BQP if and only if there exists a polynomial-time uniform family of quantum circuits, such that

In other words, a problem X is NP-easy if and only if there exists some problem Y in NP such that X is polynomial-time Turing reducible to Y.

A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it.

A decision problem is EXPSPACE-complete if it is in EXPSPACE, and every problem in EXPSPACE has a polynomial-time many-one reduction to it.

Conversely, if the Turing Machine is expected polynomial-time ( for any given x ), then a considerable fraction of the runs must be polynomial-time bounded, and the coin sequence used in such a run will be a witness.

* In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction.