armiwiki (/wiki/Main_Page)

Contents

1 Problems

2 Problems

3 Complete Problems

4 Hard Problems

4.1 Note

NP, NP Complete, NP Hard

It might be because of the name but many graduate students find it difficult to understand problems. So,I thought of explaining them in an easy way. (When explanation becomes simple, some points may be lost. So,please do refer standard text books for more information)

Problems

As the name says these problems can be solved in polynomial time, i.e.; , or , where

is a constant.

Problems

Some think as Non-Polynomial. But actually its Non-deterministic Polynomial time. i.e.; these problemscan be solved in polynomial time by a non-deterministic Turing machine and hence in exponential time by adeterministic Turing machine. In other words these problems can be verified (if a solution is given, say if itscorrect or wrong) in polynomial time. Examples include all P problems. One example of a problem not in butin is Integer Factorization (https://en.wikipedia.org/wiki/Integer_factorization_problem).

Complete Problems

Over the years many problems in have been proved to be in (like Primality Testing(https://en.wikipedia.org/wiki/Primality_test)). Still, there are many problems in not proved to be in .i.e.; the question still remains whether (i.e.; whether all problems are actually problems).

Complete Problems helps in solving the above question. They are a subset of problems with theproperty that all other problems can be reduced to any of them in polynomial time. So, they are thehardest problems in , in terms of running time. If it can be showed that any Problem is in , thenall problems in will be in (because of definition), and hence .

Search

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All problems are in (again, due to definition). Examples of problems(https://en.wikipedia.org/wiki/List_of_NP-complete_problems)

Hard Problems

These problems need not have any bound on their running time. If any Problem is polynomial timereducible to a problem , that problem belongs to Hard class. Hence, all Complete problemsare also . In other words if a problem is non-deterministic polynomial time solvable, its a

problem. Example of a problem that's not is Halting Problem(https://en.wikipedia.org/wiki/Halting_problem).

From the diagram, its clear that problems are the hardest problems in while being the simplestones in . i.e.;

Note

Given a general problem, we can say its in , if and only if we can reduce it to some problem(which shows its in NP) and also some problem can be reduced to it (which shows all NP problems canbe reduced to this problem).

Also, if a problem is in , then it's

Some Reduction Inferences (/wiki/Some_Reduction_Inferences)

--Arjun (/wiki/User:Arjun) (talk (/wiki/User_talk:Arjun)) 22:48, 16 November 2013 (UTC)