NP-complete - définition. Qu'est-ce que NP-complete
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Qu'est-ce (qui) est NP-complete - définition

PROPERTY OF COMPUTATIONAL PROBLEMS THAT IS A SPECIAL CASE OF NP-COMPLETENESS
Strongly NP-hard; Strongly NP-complete

NP-complete         
  • Levin]] proved that each easy-to-verify problem can be solved as fast as SAT, which is hence NP-complete.
  • P≠NP]], while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete, and in general, not every problem in P or NP is NP-complete)
  • reductions]] typically used to prove their NP-completeness
COMPLEXITY CLASS
NP-complete problem; NP-complete problems; NP complete; NP completeness; NP-C; Np complete; Np-complete; NP-complete language; Np-complete problem; NP-Completeness; Np completeness; Non-deterministic polynomial-time complete; NP-Complete; Nondeterministic Polynomial Complete; Non polynomial complete; Np-Complete; NP-complete; NP-incomplete
<complexity> (NPC, Nondeterministic Polynomial time complete) A set or property of computational decision problems which is a subset of NP (i.e. can be solved by a nondeterministic Turing Machine in polynomial time), with the additional property that it is also NP-hard. Thus a solution for one NP-complete problem would solve all problems in NP. Many (but not all) naturally arising problems in class NP are in fact NP-complete. There is always a polynomial-time algorithm for transforming an instance of any NP-complete problem into an instance of any other NP-complete problem. So if you could solve one you could solve any other by transforming it to the solved one. The first problem ever shown to be NP-complete was the satisfiability problem. Another example is {Hamilton's problem}. See also computational complexity, halting problem, Co-NP, NP-hard. NP/">http://fi-www.arc.nasa.gov/fia/projects/bayes-group/group/NP/. [Other examples?] (1995-04-10)
Weak NP-completeness         
SET OF COMPUTATIONAL PROBLEMS FOR WHICH THERE IS AN ALGORITHM SOLVING THEM IN POLYNOMIAL TIME IN THE DIMENSION OF THE PROBLEM AND THE MAGNITUDES OF THE DATA INVOLVED (IF GIVEN AS INTEGERS), RATHER THAN THE BASE-TWO LOGARITHMS OF THEIR MAGNITUDES
Weakly NP-complete; Weakly NP-hard
In computational complexity, an NP-complete (or NP-hard) problem is weakly NP-complete (or weakly NP-hard) if there is an algorithm for the problem whose running time is polynomial in the dimension of the problem and the magnitudes of the data involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms are technically exponential functions of their input size and are therefore not considered polynomial.
Strong NP-completeness         
In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters.

Wikipédia

Strong NP-completeness

In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters.

A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a polynomial reduction to it; in combinatorial optimization, particularly, the phrase "strongly NP-hard" is reserved for problems that are not known to have a polynomial reduction to another strongly NP-complete problem.

Normally numerical parameters to a problem are given in positional notation, so a problem of input size n might contain parameters whose size is exponential in n. If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size. Thus strong NP-completeness or NP-hardness may also be defined as the NP-completeness or NP-hardness of this unary version of the problem.

For example, bin packing is strongly NP-complete while the 0-1 Knapsack problem is only weakly NP-complete. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming.

From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP. However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded.

Some strongly NP-complete problems may still be easy to solve on average, but it's more likely that difficult instances will be encountered in practice.