Arjun Suresh (talk | contribs) (→NP Problems) |
|||
| (129 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| − | + | <metadesc>NP Complete definition in simple terms. Differences between P, NP, NP Complete and NP Hard Problems</metadesc> | |
| − | |||
| − | |||
| − | + | It might be because of the name but many graduate students find it difficult to understand <math>NP</math> 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) | |
| − | |||
| − | |||
| − | == | + | ==<math>P</math> Problems== |
| − | + | As the name says these problems can be solved in polynomial time, i.e.; <math>O(n)</math>, <math>O(n^2)</math> or <math>O(n^k)</math>, where <math>k</math> is a constant. | |
| − | <math>NP</math> | + | ==<math>NP</math> Problems== |
| − | + | Some think <math>NP</math> as Non-Polynomial. But actually it is [https://en.wikipedia.org/wiki/NP_(complexity) Non-deterministic Polynomial time]. i.e.; "yes" instances of these problems can be solved in polynomial time by a non-deterministic Turing machine and hence can take up to exponential time (some problems can be solved in sub-exponential but super polynomial time) by a deterministic Turing machine. In other words these problems can be verified (if a solution is given, say if it is correct or wrong) in polynomial time. Examples include all P problems. One example of a problem not in <math>P</math> but in <math>NP</math> is [https://en.wikipedia.org/wiki/Integer_factorization_problem Integer Factorization]. | |
| − | ==NP | + | ==<math>NP</math> Complete Problems<math>(NPC)</math>== |
| − | + | Over the years many problems in <math>NP</math> have been proved to be in <math>P</math> (like [https://en.wikipedia.org/wiki/Primality_test Primality Testing]). Still, there are many problems in <math>NP</math> not proved to be in <math>P</math>. i.e.; the question still remains whether <math>P = NP</math> (i.e.; whether all <math>NP</math> problems are actually <math>P</math> problems). | |
| − | [[Category: | + | <math>NP</math> Complete Problems helps in solving the above question. They are a subset of <math>NP</math> problems with the property that all other <math>NP</math> problems can be reduced to any of them in polynomial time. So, they are the hardest problems in <math>NP</math>, in terms of running time. If it can be showed that any <math>NPC</math> Problem is in <math>P</math>, then all problems in <math>NP</math> will be in <math>P</math> (because of <math>NPC</math> definition), and hence <math>P = NP = NPC</math>. |
| + | |||
| + | All <math>NPC</math> problems are in <math>NP</math> (again, due to <math>NPC</math> definition). Examples of [https://en.wikipedia.org/wiki/List_of_NP-complete_problems <math>NPC</math> problems] | ||
| + | |||
| + | ==<math>NP</math> Hard Problems <math>(NPH)</math>== | ||
| + | These problems need not have any bound on their running time. If any <math>NPC</math> Problem is polynomial time reducible to a problem <math>X</math>, that problem <math>X</math> belongs to <math>NP</math> Hard class. Hence, all <math>NP</math> Complete problems are also <math>NPH</math>. In other words if a <math>NPH</math> problem is non-deterministic polynomial time solvable, its a <math>NPC</math> problem. Example of a <math>NP</math> problem that's not <math>NPC</math> is [https://en.wikipedia.org/wiki/Halting_problem Halting Problem]. | ||
| + | |||
| + | |||
| + | https://upload.wikimedia.org/wikipedia/commons/thumb/a/a0/P_np_np-complete_np-hard.svg/400px-P_np_np-complete_np-hard.svg.png | ||
| + | |||
| + | |||
| + | From the diagram, its clear that <math>NPC</math> problems are the hardest problems in <math>NP</math> while being the simplest ones in <math>NPH</math>. i.e.; | ||
| + | $NP ∩ NPH = NPC$ | ||
| + | |||
| + | ===Note=== | ||
| + | Given a general problem, we can say its in <math>NPC</math>, if and only if we can reduce it to some <math>NP</math> problem (which shows its in NP) and also some <math>NPC</math> problem can be reduced to it (which shows all NP problems can be reduced to this problem). | ||
| + | |||
| + | Also, if a <math>NPH</math> problem is in <math>NP</math>, then it's <math>NPC</math> | ||
| + | |||
| + | [[Some Reduction Inferences|Some Reduction Inferences ]] | ||
| + | |||
| + | --[[User:Arjun|Arjun]] ([[User talk:Arjun|talk]]) 22:48, 16 November 2013 (UTC) | ||
| + | |||
| + | {{Template:FBD}} | ||
| + | |||
| + | |||
| + | [[Category: Automata Theory Notes]] | ||
| + | |||
| + | |||
| + | [[Category: Compact Notes for Reference of Understanding]] | ||
It might be because of the name but many graduate students find it difficult to understand <math>NP</math> 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)
As the name says these problems can be solved in polynomial time, i.e.; <math>O(n)</math>, <math>O(n^2)</math> or <math>O(n^k)</math>, where <math>k</math> is a constant.
Some think <math>NP</math> as Non-Polynomial. But actually it is Non-deterministic Polynomial time. i.e.; "yes" instances of these problems can be solved in polynomial time by a non-deterministic Turing machine and hence can take up to exponential time (some problems can be solved in sub-exponential but super polynomial time) by a deterministic Turing machine. In other words these problems can be verified (if a solution is given, say if it is correct or wrong) in polynomial time. Examples include all P problems. One example of a problem not in <math>P</math> but in <math>NP</math> is Integer Factorization.
Over the years many problems in <math>NP</math> have been proved to be in <math>P</math> (like Primality Testing). Still, there are many problems in <math>NP</math> not proved to be in <math>P</math>. i.e.; the question still remains whether <math>P = NP</math> (i.e.; whether all <math>NP</math> problems are actually <math>P</math> problems).
<math>NP</math> Complete Problems helps in solving the above question. They are a subset of <math>NP</math> problems with the property that all other <math>NP</math> problems can be reduced to any of them in polynomial time. So, they are the hardest problems in <math>NP</math>, in terms of running time. If it can be showed that any <math>NPC</math> Problem is in <math>P</math>, then all problems in <math>NP</math> will be in <math>P</math> (because of <math>NPC</math> definition), and hence <math>P = NP = NPC</math>.
All <math>NPC</math> problems are in <math>NP</math> (again, due to <math>NPC</math> definition). Examples of <math>NPC</math> problems
These problems need not have any bound on their running time. If any <math>NPC</math> Problem is polynomial time reducible to a problem <math>X</math>, that problem <math>X</math> belongs to <math>NP</math> Hard class. Hence, all <math>NP</math> Complete problems are also <math>NPH</math>. In other words if a <math>NPH</math> problem is non-deterministic polynomial time solvable, its a <math>NPC</math> problem. Example of a <math>NP</math> problem that's not <math>NPC</math> is Halting Problem.
From the diagram, its clear that <math>NPC</math> problems are the hardest problems in <math>NP</math> while being the simplest ones in <math>NPH</math>. i.e.;
$NP ∩ NPH = NPC$
Given a general problem, we can say its in <math>NPC</math>, if and only if we can reduce it to some <math>NP</math> problem (which shows its in NP) and also some <math>NPC</math> problem can be reduced to it (which shows all NP problems can be reduced to this problem).
Also, if a <math>NPH</math> problem is in <math>NP</math>, then it's <math>NPC</math>
--Arjun (talk) 22:48, 16 November 2013 (UTC)
It might be because of the name that many graduate students find it difficult to understand about NP problems. So, I thought of explaining them as easy as I can.
As the name says these problems can be solved in polynomial time, i.e.; <math>O(n), O(n^2) or O(n^k), </math>where <math>k</math> is a constant.
<math>\sigma</math> Some think <math>NP</math> as Non-Polynomial. But actually its Non-deterministic Polynomial time. i.e.; these problems can be solved in polynomial time by a non-deterministic Turing machine and hence in exponential time by a deterministic Turing machine. In other words these problems can be verified (if a solution is given, say if its correct or wrong) in polynomial time. Examples include all P problems and one example of a problem not in <math>P</math> but in <math>NP</math> is Integer Factorization.
$O$ver the years many problems in <math>NP</math> have been proved to be in <math>P</math> also (like Primality Testing). Still, there are many problems in <math>NP</math> still not proved to be in <math>P</math>. i.e.; the question still remains whether <math>P = NP</math>.
<math>NP</math> Complete Problems are an aid in solving this question. They are a subset of <math>NP</math> problems with the property that all other <math>NP</math> problems can be reduced to any <math>NP</math> Complete problem in polynomial time. So, they are the hardest problems in <math>NP</math> in terms of running time. Also, if anyone can show that an <math>NP</math> Complete Problem is in <math>P</math>, then all problems in <math>NP</math> will be in <math>P</math>, and hence <math>P = NP = NPC</math>.
All <math>NP</math> Complete problems are in <math>NP</math> because of the definition. Examples of <math>NP</math> Complete problems
These problems need not have any bound on their running time. If any NP Complete Problem is polynomial time reducable to a problem, that problem belongs to NP Hard class. Hence, all NP Complete problems are also NP Hard. In other words if a NP Hard problem is non-deterministic polynomial time solvable, its a NP Complete problem. Example of a NP Hard problem that's not NP Complete is Halting Problem.
GATECSE