Question what is the relationship between

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Question what is the relationship between

However, the best known quantum algorithm for this problem, Shor's algorithmdoes run in polynomial time, although this does not indicate where the problem lies with respect to non-quantum complexity classes. Does P mean "easy"?

Quadratic fit suggests that empirical algorithmic complexity for instances with 50—10, variables is O log n 2. It is a common and reasonably accurate assumption in complexity theory; however, it has some caveats.

First, it is not always true in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents thus rendering it impractical. There are algorithms for many NP-complete problems, such as the knapsack problemthe traveling salesman problem and the Boolean satisfiability problemthat can solve to optimality many real-world instances in reasonable time.

The empirical average-case complexity time vs. An example is the simplex algorithm in linear programmingwhich works surprisingly well in practice; despite having exponential worst-case time complexity it runs on par with the best known polynomial-time algorithms.

A key reason for this belief is that after decades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than important known NP-complete problems see List of NP-complete problems. These algorithms were sought long before the concept of NP-completeness was even defined Karp's 21 NP-complete problemsamong the first found, were all well-known existing problems at the time they were shown to be NP-complete.

It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easy to verify matches real-world experience.

There would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found.

For example, in these statements were made: This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration.

VardiRice University Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required.

Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well. It is also possible that a proof would not lead directly to efficient methods, perhaps if the proof is non-constructiveor the size of the bounding polynomial is too big to be efficient in practice.

The consequences, both positive and negative, arise since various NP-complete problems are fundamental in many fields. Cryptography, for example, relies on certain problems being difficult.

A constructive and efficient solution [Note 2] to an NP-complete problem such as 3-SAT would break most existing cryptosystems including: Existing implementations of public-key cryptography[27] a foundation for many modern security applications such as secure financial transactions over the Internet.

Question what is the relationship between

Cryptographic hashing as the problem of finding a pre-image that hashes to a given value must be difficult in order to be useful, and ideally should require exponential time. On the other hand, there are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems.

For instance, many problems in operations research are NP-complete, such as some types of integer programming and the travelling salesman problem.

Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in protein structure predictionare also NP-complete; [30] if these problems were efficiently solvable it could spur considerable advances in life sciences and biotechnology.

But such changes may pale in significance compared to the revolution an efficient method for solving NP-complete problems would cause in mathematics itself. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblemthe mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine.

After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem. Similarly, Stephen Cook says [33] Example problems may well include all of the CMI prize problems.

Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries to prove.

A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, would essentially end this struggle. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems.

For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable.

Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average-case complexity question. A Princeton University workshop in studied the status of the five worlds.Delegation strategies for the NCLEX, Prioritization for the NCLEX, Infection Control for the NCLEX, FREE resources for the NCLEX, FREE NCLEX Quizzes for the NCLEX, FREE NCLEX exams for the NCLEX, Failed the NCLEX - Help is here.

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