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What if P vs NP is solved?

What if P vs NP is solved?

If P equals NP, every NP problem would contain a hidden shortcut, allowing computers to quickly find perfect solutions to them. But if P does not equal NP, then no such shortcuts exist, and computers’ problem-solving powers will remain fundamentally and permanently limited.

What is the relation between P and NP class problems is P NP If no then what will happen if P will become equal to NP?

There are a large number of important problems that are known to be NP-complete (basically, if any these problems are proven to be in P, then all NP problems are proven to be in P). If P = NP, then all of these problems will be proven to have an efficient (polynomial time) solution. Most scientists believe that P!= NP.

What happens when P will be equal to NP?

If P=NP, then all of the NP problems can be solved deterministically in Polynomial time. This is because the NP problems are all essentially the same problem, just stated in different terms.

Can a problem be P and NP-complete?

A problem p in NP is NP-complete if every other problem in NP can be transformed (or reduced) into p in polynomial time. It is not known whether every problem in NP can be quickly solved—this is called the P versus NP problem.

Why does P vs NP matter?

Roughly speaking, P is a set of relatively easy problems, and NP is a set that includes what seem to be very, very hard problems, so P = NP would imply that the apparently hard problems actually have relatively easy solutions.

Why is P NP so important?

Now, if P=NP, we could find solutions to search problems as easily as checking whether those solutions are good. This would essentially solve all the algorithmic challenges that we face today and computers could solve almost any task.

How do you prove a problem is NP-hard?

To prove that problem A is NP-hard, reduce a known NP-hard problem to A. In other words, to prove that your problem is hard, you need to describe an ecient algorithm to solve a dierent problem, which you already know is hard, using an hypothetical ecient algorithm for your problem as a black-box subroutine.

What is NP-hard problem with example?

An example of an NP-hard problem is the decision subset sum problem: given a set of integers, does any non-empty subset of them add up to zero? That is a decision problem and happens to be NP-complete.

Has anyone solved NP or P?

Although one-way functions have never been formally proven to exist, most mathematicians believe that they do, and a proof of their existence would be a much stronger statement than P ≠ NP. Thus it is unlikely that natural proofs alone can resolve P = NP.

Why is P vs NP so hard?

According to polls, most computer scientists believe that P ≠ NP. 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 3000 important known NP-complete problems (see List of NP-complete problems).

Is P vs NP solvable?

P is the set of all decision problems that are efficiently solvable. P is a subset of NP. P is the set of all decision problems that are efficiently solvable and is a subset of NP. Basic Arithmetic is solvable in Polynomial-time, thus belongs to P.

Is the P NP problem solved?

Thus if any one NP-Complete problem can be solved in polynomial time, then every NP-Complete problem can be solved in polynomial time, and every problem in NP can be solved in polynomial time (i.e. P=NP).

What is the P versus NP problem in Computer Science?

The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified (technically, verified in polynomial time) can also be solved quickly (again, in polynomial time).

How is NP completeness used to attack the P question?

To attack the P = NP question, the concept of NP -completeness is very useful. NP -complete problems are a set of problems to each of which any other NP -problem can be reduced in polynomial time and whose solution may still be verified in polynomial time. That is, any NP problem can be transformed into any of the NP -complete problems.

Which is the complexity class P or NP?

The million-dollar question is whether the two complexity classes P and NP equal each other. P is contained in NP: Any problem that can be solved quickly by a computer can also have a particular possible answer quickly checked by a computer.

When did John Nash come up with the P vs NP problem?

In 1955, mathematician John Nash wrote a letter to the NSA, where he speculated that cracking a sufficiently complex code would require time exponential in the length of the key. If proved (and Nash was suitably skeptical) this would imply what is now called P ≠ NP, since a proposed key can easily be verified in polynomial time.