What is an NP-complete in computer science

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What is an NP-complete problem  Why is it such an important topic in computer science

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What is NP  NP is the set of all decision problems  questions with a yes-or-no answer  for which the  yes -answers can be verified in polynomial time  O nk  where n is the problem size  and k is a constant  by a deterministic Turing machine  Polynomial time is sometimes used as the definition of fast or quickly  What is P  P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine  Since they can be solved in polynomial time  they can also be verified in polynomial time  Therefore P is a subset of NP  What is NP-Complete  A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly  ie  in polynomial time  transformed into x  In other words   x is in NP  and Every problem in NP is reducible to x  So  what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly  then all NP problems can be solved quickly  See also the post What s  quot P NP  quot   and why is it such a famous question  What is NP-Hard  NP-Hard are problems that are at least as hard as the hardest problems in NP  Note that NP-Complete problems are also NP-hard  However not all NP-hard problems are NP  or even a decision problem   despite having NP as a prefix  That is the NP in NP-hard does not mean non-deterministic polynomial time  Yes  this is confusing  but its usage is entrenched and unlikely to change

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NP stands for Non-deterministic Polynomial time   This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine  like a regular Turing machine but also including a non-deterministic  choice  function   Basically  a solution has to be testable in poly time  If that s the case  and a known NP problem can be solved using the given problem with modified input  an NP problem can be reduced to the given problem  then the problem is NP complete   The main thing to take away from an NP-complete problem is that it cannot be solved in polynomial time in any known way  NP-Hard NP-Complete is a way of showing that certain classes of problems are not solvable in realistic time   Edit  As others have noted  there are often approximate solutions for NP-Complete problems  In this case  the approximate solution usually gives an approximation bound using special notation which tells us how close the approximation is

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NP stands for Non-deterministic Polynomial time   This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine  like a regular Turing machine but also including a non-deterministic  choice  function   Basically  a solution has to be testable in poly time  If that s the case  and a known NP problem can be solved using the given problem with modified input  an NP problem can be reduced to the given problem  then the problem is NP complete   The main thing to take away from an NP-complete problem is that it cannot be solved in polynomial time in any known way  NP-Hard NP-Complete is a way of showing that certain classes of problems are not solvable in realistic time   Edit  As others have noted  there are often approximate solutions for NP-Complete problems  In this case  the approximate solution usually gives an approximation bound using special notation which tells us how close the approximation is

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NP-Complete is a class of problems    The class P consists of those problems that are solvable in polynomial time  For example  they could be solved in O nk  for some constant k  where n is the size of the input  Simply put  you can write a program that will run in reasonable time   The class NP consists of those problems that are verifiable in polynomial time  That is  if we are given a potential solution  then we could check if the given solution is correct in polynomial time   Some examples are the Boolean Satisfiability  or SAT  problem  or the Hamiltonian-cycle problem  There are many problems that are known to be in the class NP    NP-Complete means the problem is at least as hard as any problem in NP   It is important to computer science because it has been proven that any problem in NP can be transformed into another problem in NP-complete  That means that a solution to any one NP-complete problem is a solution to all NP problems   Many algorithms in security depends on the fact that no known solutions exist for NP hard problems  It would definitely have a significant impact on computing if a solution were found

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NP-Complete means something very specific and you have to be careful or you will get the definition wrong   First  an NP problem is a yes no problem such that   There is polynomial-time proof for every instance of the problem with a  yes  answer that the answer is  yes   or  equivalently  There exists a polynomial-time algorithm  possibly using random variables  that has a non-zero probability of answering  yes  if the answer to an instance of the problem is  yes  and will say  no  100  of the time if the answer is  no   In other words  the algorithm must have a false-negative rate less than 100  and no false positives    A problem X is NP-Complete if   X is in NP  and For any problem Y in NP  there is a  reduction  from Y to X  a polynomial-time algorithm that transforms any instance of Y into an instance of X such that the answer to the Y-instance is  yes  if and only if the answer X-instance is  yes     If X is NP-complete and a deterministic  polynomial-time algorithm exists that can solve all instances of X correctly  0  false-positives  0  false-negatives   then any problem in NP can be solved in deterministic-polynomial-time  by reduction to X    So far  nobody has come up with such a deterministic polynomial-time algorithm  but nobody has proven one doesn t exist  there s a million bucks for anyone who can do either  the is the P   NP problem    That doesn t mean that you can t solve a particular instance of an NP-Complete  or NP-Hard  problem   It just means you can t have something that will work reliably on all instances of a problem the same way you could reliably sort a list of integers   You might very well be able to come up with an algorithm that will work very well on all practical instances of a NP-Hard problem

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If you re looking for an example of an NP-complete problem then I suggest you take a look at 3-SAT   The basic premise is you have an expression in conjunctive normal form  which is a way of saying you have a series of expressions joined by ORs that all must be true    a or b  and  b or  c  and  d or  e or f        The 3-SAT problem is to find a solution that will satisfy the expression where each of the OR-expressions has exactly 3 booleans to match    a or  b or  c  and   a or b or  d  and  b or  c or d        A solution to this one might be  a T  b T  c F  d F   However  no algorithm has been discovered that will solve this problem in the general case in polynomial time  What this means is that the best way to solve this problem is to do essentially a brute force guess-and-check and try different combinations until you find one that works   What s special about the 3-SAT problem is that ANY NP-complete problem can be reduced to a 3-SAT problem  This means that if you can find a polynomial-time algorithm to solve this problem then you get  1 000 000  not to mention the respect and admiration of computer scientists and mathematicians around the world

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NP-Complete is a class of problems    The class P consists of those problems that are solvable in polynomial time  For example  they could be solved in O nk  for some constant k  where n is the size of the input  Simply put  you can write a program that will run in reasonable time   The class NP consists of those problems that are verifiable in polynomial time  That is  if we are given a potential solution  then we could check if the given solution is correct in polynomial time   Some examples are the Boolean Satisfiability  or SAT  problem  or the Hamiltonian-cycle problem  There are many problems that are known to be in the class NP    NP-Complete means the problem is at least as hard as any problem in NP   It is important to computer science because it has been proven that any problem in NP can be transformed into another problem in NP-complete  That means that a solution to any one NP-complete problem is a solution to all NP problems   Many algorithms in security depends on the fact that no known solutions exist for NP hard problems  It would definitely have a significant impact on computing if a solution were found

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I have heard an explanation  that is   NP-Completeness is probably one of the more enigmatic ideas in the study of algorithms   NP  stands for  nondeterministic polynomial time   and is the name for what is called a complexity class to which problems can belong  The important thing about the NP complexity class is that problems within that class can be verified by a polynomial time algorithm  As an example  consider the problem of counting stuff  Suppose there are a bunch of apples on a table  The problem is  How many apples are there   You are provided with a possible answer  8  You can verify this answer in polynomial time by using the algorithm of  duh  counting the apples  Counting the apples happens in O n   that s Big-oh notation  time  because it takes one step to count each apple  For n apples  you need n steps  This problem is in the NP complexity class   A problem is classified as NP-complete if it can be shown that it is both NP-Hard and verifiable in polynomial time  Without going too deeply into the discussion of NP-Hard  suffice it to say that there are certain problems to which polynomial time solutions have not been found  That is  it takes something like n   n factorial  steps to solve them  However  if you re given a solution to an NP-Complete problem  you can verify it in polynomial time   A classic example of an NP-Complete problem is The Traveling Salesman Problem    The author  ApoxyButt From  http   www everything2 com title NP-complete

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The definitions for NP complete problems above is correct  but I thought I might wax lyrical about their philosophical importance as nobody has addressed that issue yet   Almost all complex problems you ll come up against will be NP Complete  There s something very fundamental about this class  and which just seems to be computationally different from easily solvable problems  They sort of have their own flavour  and it s not so hard to recognise them  This basically means that any moderately complex algorithm is impossible for you to solve exactly -- scheduling  optimising  packing  covering etc   But not all is lost if a problem you ll encounter is NP Complete  There is a vast and very technical field where people study approximation algorithms  which will give you guarantees for being close to the solution of an NP complete problem  Some of these are incredibly strong guarantees -- for example  for 3sat  you can get a 7 8 guarantee through a really obvious algorithm  Even better  in reality  there are some very strong heuristics  which excel at giving great answers  but no guarantees   for these problems   Note that two very famous problems -- graph isomorphism and factoring -- are not known to be P or NP

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The definitions for NP complete problems above is correct  but I thought I might wax lyrical about their philosophical importance as nobody has addressed that issue yet   Almost all complex problems you ll come up against will be NP Complete  There s something very fundamental about this class  and which just seems to be computationally different from easily solvable problems  They sort of have their own flavour  and it s not so hard to recognise them  This basically means that any moderately complex algorithm is impossible for you to solve exactly -- scheduling  optimising  packing  covering etc   But not all is lost if a problem you ll encounter is NP Complete  There is a vast and very technical field where people study approximation algorithms  which will give you guarantees for being close to the solution of an NP complete problem  Some of these are incredibly strong guarantees -- for example  for 3sat  you can get a 7 8 guarantee through a really obvious algorithm  Even better  in reality  there are some very strong heuristics  which excel at giving great answers  but no guarantees   for these problems   Note that two very famous problems -- graph isomorphism and factoring -- are not known to be P or NP

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What is NP  NP is the set of all decision problems  questions with a yes-or-no answer  for which the  yes -answers can be verified in polynomial time  O nk  where n is the problem size  and k is a constant  by a deterministic Turing machine  Polynomial time is sometimes used as the definition of fast or quickly  What is P  P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine  Since they can be solved in polynomial time  they can also be verified in polynomial time  Therefore P is a subset of NP  What is NP-Complete  A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly  ie  in polynomial time  transformed into x  In other words   x is in NP  and Every problem in NP is reducible to x  So  what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly  then all NP problems can be solved quickly  See also the post What s  quot P NP  quot   and why is it such a famous question  What is NP-Hard  NP-Hard are problems that are at least as hard as the hardest problems in NP  Note that NP-Complete problems are also NP-hard  However not all NP-hard problems are NP  or even a decision problem   despite having NP as a prefix  That is the NP in NP-hard does not mean non-deterministic polynomial time  Yes  this is confusing  but its usage is entrenched and unlikely to change

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NP Problem  -  NP problem are such problem that can be solved in non-deterministic polynomial time  Non deterministic algorithm operate in two stage  Non deterministic guessing stage  amp  amp  Non deterministic verification stage   Type of Np Problem  NP complete NP Hard  NP Complete problem  - 1 Decision Problem A is called NP complete if it has following two properties -  It belong to class NP  Every other problem in NP can be transformed to P in polynomial time   Some Ex  -  Knapsack problem sub set sum problem Vertex covering problem

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Honestly  Wikipedia might be the best place to look for an answer to this   If NP   P  then we can solve very hard problems much faster than we thought we could before   If we solve only one NP-Complete problem in P  polynomial  time  then it can be applied to all other problems in the NP-Complete category

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The definitions for NP complete problems above is correct  but I thought I might wax lyrical about their philosophical importance as nobody has addressed that issue yet   Almost all complex problems you ll come up against will be NP Complete  There s something very fundamental about this class  and which just seems to be computationally different from easily solvable problems  They sort of have their own flavour  and it s not so hard to recognise them  This basically means that any moderately complex algorithm is impossible for you to solve exactly -- scheduling  optimising  packing  covering etc   But not all is lost if a problem you ll encounter is NP Complete  There is a vast and very technical field where people study approximation algorithms  which will give you guarantees for being close to the solution of an NP complete problem  Some of these are incredibly strong guarantees -- for example  for 3sat  you can get a 7 8 guarantee through a really obvious algorithm  Even better  in reality  there are some very strong heuristics  which excel at giving great answers  but no guarantees   for these problems   Note that two very famous problems -- graph isomorphism and factoring -- are not known to be P or NP

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It s a class of problems where we must simulate every possibility to be sure we have the optimal solution    There are a lot of good heuristics for some NP-Complete problems  but they are only an educated guess at best

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As far as I understand P is the set of problems which could be solved in polynomial time with a deterministic TM  NP is the set of problems which requires a non-deterministic TM to be solved in polynomial time  This means that we can check all different combinations of variables in parallel with each instance taking polynomial time  If the problem is solvable then at least one of those parallel instances of TM would halt with  quot yes quot   This also means that if you could make a correct guess about the variables solution then you just need to check it s validity in polynomial time  NP-Hard is the set where problems are at least as hard as NP  This means NP-Hard problem are equally or more difficult than any problem in NP set   Example of a NP-hard problem which is more difficult than NP is the halting problem  Example of a NP-hard problem which is equally difficult to a NP problem is any problem in NP-Complete set  see below    NP-Complete is the intersection set of NP and NP-Hard  A problems in NP-Complete set can be reduced to any other NP-Complete problem  That means if any of the NP-Complete problem would have an efficient solution then all of the NP-Complete problems could be solved with same solution  How can you prove P NP and win a US  1 000 000 prize  If you could show that every NP problem is reducible to any other NP problem  then NP-Complete set spans over entire NP sets  definition of NP-Complete   i e NP-Complete NP  Also since P NP  this means that at least one of those NP-Complete problem is solvable in polynomial time  which implies that entire NP set can be solved in polynomial time  Clearly it would conclude that P NP  Well this is just one way to prove P NP  there might be many other ways  Please let me know if I made any mistake

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What is NP  NP is the set of all decision problems  questions with a yes-or-no answer  for which the  yes -answers can be verified in polynomial time  O nk  where n is the problem size  and k is a constant  by a deterministic Turing machine  Polynomial time is sometimes used as the definition of fast or quickly  What is P  P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine  Since they can be solved in polynomial time  they can also be verified in polynomial time  Therefore P is a subset of NP  What is NP-Complete  A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly  ie  in polynomial time  transformed into x  In other words   x is in NP  and Every problem in NP is reducible to x  So  what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly  then all NP problems can be solved quickly  See also the post What s  quot P NP  quot   and why is it such a famous question  What is NP-Hard  NP-Hard are problems that are at least as hard as the hardest problems in NP  Note that NP-Complete problems are also NP-hard  However not all NP-hard problems are NP  or even a decision problem   despite having NP as a prefix  That is the NP in NP-hard does not mean non-deterministic polynomial time  Yes  this is confusing  but its usage is entrenched and unlikely to change

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We need to separate algorithms and problems  We write algorithms to solve problems  and they scale in a certain way  Although this is a simplification  let s label an algorithm with a  P  if the scaling is good enough  and  NP  if it isn t   It s helpful to know things about the problems we re trying to solve  rather than the algorithms we use to solve them  So we ll say that all the problems which have a well-scaling algorithm are  in P   And the ones which have a poor-scaling algorithm are  in NP    That means that lots of simple problems are  in NP  too  because we can write bad algorithms to solve easy problems  It would be good to know which problems in NP are the really tricky ones  but we don t just want to say  it s the ones we haven t found a good algorithm for   After all  I could come up with a problem  call it X  that I think needs a super-amazing algorithm  I tell the world that the best algorithm I could come up with to solve X scales badly  and so I think that X is a really tough problem  But tomorrow  maybe somebody cleverer than me invents an algorithm which solves X and is in P  So this isn t a very good definition of hard problems   All the same  there are lots of problems in NP that nobody knows a good algorithm for  So if I could prove that X is a certain sort of problem  one where a good algorithm to solve X could also be used  in some roundabout way  to give a good algorithm for every other problem in NP  Well now people might be a bit more convinced that X is a genuinely tricky problem  And in this case we call X NP-Complete

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We need to separate algorithms and problems  We write algorithms to solve problems  and they scale in a certain way  Although this is a simplification  let s label an algorithm with a  P  if the scaling is good enough  and  NP  if it isn t   It s helpful to know things about the problems we re trying to solve  rather than the algorithms we use to solve them  So we ll say that all the problems which have a well-scaling algorithm are  in P   And the ones which have a poor-scaling algorithm are  in NP    That means that lots of simple problems are  in NP  too  because we can write bad algorithms to solve easy problems  It would be good to know which problems in NP are the really tricky ones  but we don t just want to say  it s the ones we haven t found a good algorithm for   After all  I could come up with a problem  call it X  that I think needs a super-amazing algorithm  I tell the world that the best algorithm I could come up with to solve X scales badly  and so I think that X is a really tough problem  But tomorrow  maybe somebody cleverer than me invents an algorithm which solves X and is in P  So this isn t a very good definition of hard problems   All the same  there are lots of problems in NP that nobody knows a good algorithm for  So if I could prove that X is a certain sort of problem  one where a good algorithm to solve X could also be used  in some roundabout way  to give a good algorithm for every other problem in NP  Well now people might be a bit more convinced that X is a genuinely tricky problem  And in this case we call X NP-Complete

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Honestly  Wikipedia might be the best place to look for an answer to this   If NP   P  then we can solve very hard problems much faster than we thought we could before   If we solve only one NP-Complete problem in P  polynomial  time  then it can be applied to all other problems in the NP-Complete category

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If you re looking for an example of an NP-complete problem then I suggest you take a look at 3-SAT   The basic premise is you have an expression in conjunctive normal form  which is a way of saying you have a series of expressions joined by ORs that all must be true    a or b  and  b or  c  and  d or  e or f        The 3-SAT problem is to find a solution that will satisfy the expression where each of the OR-expressions has exactly 3 booleans to match    a or  b or  c  and   a or b or  d  and  b or  c or d        A solution to this one might be  a T  b T  c F  d F   However  no algorithm has been discovered that will solve this problem in the general case in polynomial time  What this means is that the best way to solve this problem is to do essentially a brute force guess-and-check and try different combinations until you find one that works   What s special about the 3-SAT problem is that ANY NP-complete problem can be reduced to a 3-SAT problem  This means that if you can find a polynomial-time algorithm to solve this problem then you get  1 000 000  not to mention the respect and admiration of computer scientists and mathematicians around the world

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NP-Complete is a class of problems    The class P consists of those problems that are solvable in polynomial time  For example  they could be solved in O nk  for some constant k  where n is the size of the input  Simply put  you can write a program that will run in reasonable time   The class NP consists of those problems that are verifiable in polynomial time  That is  if we are given a potential solution  then we could check if the given solution is correct in polynomial time   Some examples are the Boolean Satisfiability  or SAT  problem  or the Hamiltonian-cycle problem  There are many problems that are known to be in the class NP    NP-Complete means the problem is at least as hard as any problem in NP   It is important to computer science because it has been proven that any problem in NP can be transformed into another problem in NP-complete  That means that a solution to any one NP-complete problem is a solution to all NP problems   Many algorithms in security depends on the fact that no known solutions exist for NP hard problems  It would definitely have a significant impact on computing if a solution were found

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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      Informally  an NP-complete problem is an NP problem that is at least as  tough  as any other problem in NP

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I have heard an explanation  that is   NP-Completeness is probably one of the more enigmatic ideas in the study of algorithms   NP  stands for  nondeterministic polynomial time   and is the name for what is called a complexity class to which problems can belong  The important thing about the NP complexity class is that problems within that class can be verified by a polynomial time algorithm  As an example  consider the problem of counting stuff  Suppose there are a bunch of apples on a table  The problem is  How many apples are there   You are provided with a possible answer  8  You can verify this answer in polynomial time by using the algorithm of  duh  counting the apples  Counting the apples happens in O n   that s Big-oh notation  time  because it takes one step to count each apple  For n apples  you need n steps  This problem is in the NP complexity class   A problem is classified as NP-complete if it can be shown that it is both NP-Hard and verifiable in polynomial time  Without going too deeply into the discussion of NP-Hard  suffice it to say that there are certain problems to which polynomial time solutions have not been found  That is  it takes something like n   n factorial  steps to solve them  However  if you re given a solution to an NP-Complete problem  you can verify it in polynomial time   A classic example of an NP-Complete problem is The Traveling Salesman Problem    The author  ApoxyButt From  http   www everything2 com title NP-complete

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NP-Complete means something very specific and you have to be careful or you will get the definition wrong   First  an NP problem is a yes no problem such that   There is polynomial-time proof for every instance of the problem with a  yes  answer that the answer is  yes   or  equivalently  There exists a polynomial-time algorithm  possibly using random variables  that has a non-zero probability of answering  yes  if the answer to an instance of the problem is  yes  and will say  no  100  of the time if the answer is  no   In other words  the algorithm must have a false-negative rate less than 100  and no false positives    A problem X is NP-Complete if   X is in NP  and For any problem Y in NP  there is a  reduction  from Y to X  a polynomial-time algorithm that transforms any instance of Y into an instance of X such that the answer to the Y-instance is  yes  if and only if the answer X-instance is  yes     If X is NP-complete and a deterministic  polynomial-time algorithm exists that can solve all instances of X correctly  0  false-positives  0  false-negatives   then any problem in NP can be solved in deterministic-polynomial-time  by reduction to X    So far  nobody has come up with such a deterministic polynomial-time algorithm  but nobody has proven one doesn t exist  there s a million bucks for anyone who can do either  the is the P   NP problem    That doesn t mean that you can t solve a particular instance of an NP-Complete  or NP-Hard  problem   It just means you can t have something that will work reliably on all instances of a problem the same way you could reliably sort a list of integers   You might very well be able to come up with an algorithm that will work very well on all practical instances of a NP-Hard problem

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If you re looking for an example of an NP-complete problem then I suggest you take a look at 3-SAT   The basic premise is you have an expression in conjunctive normal form  which is a way of saying you have a series of expressions joined by ORs that all must be true    a or b  and  b or  c  and  d or  e or f        The 3-SAT problem is to find a solution that will satisfy the expression where each of the OR-expressions has exactly 3 booleans to match    a or  b or  c  and   a or b or  d  and  b or  c or d        A solution to this one might be  a T  b T  c F  d F   However  no algorithm has been discovered that will solve this problem in the general case in polynomial time  What this means is that the best way to solve this problem is to do essentially a brute force guess-and-check and try different combinations until you find one that works   What s special about the 3-SAT problem is that ANY NP-complete problem can be reduced to a 3-SAT problem  This means that if you can find a polynomial-time algorithm to solve this problem then you get  1 000 000  not to mention the respect and admiration of computer scientists and mathematicians around the world

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NP-Complete is a class of problems    The class P consists of those problems that are solvable in polynomial time  For example  they could be solved in O nk  for some constant k  where n is the size of the input  Simply put  you can write a program that will run in reasonable time   The class NP consists of those problems that are verifiable in polynomial time  That is  if we are given a potential solution  then we could check if the given solution is correct in polynomial time   Some examples are the Boolean Satisfiability  or SAT  problem  or the Hamiltonian-cycle problem  There are many problems that are known to be in the class NP    NP-Complete means the problem is at least as hard as any problem in NP   It is important to computer science because it has been proven that any problem in NP can be transformed into another problem in NP-complete  That means that a solution to any one NP-complete problem is a solution to all NP problems   Many algorithms in security depends on the fact that no known solutions exist for NP hard problems  It would definitely have a significant impact on computing if a solution were found

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We need to separate algorithms and problems  We write algorithms to solve problems  and they scale in a certain way  Although this is a simplification  let s label an algorithm with a  P  if the scaling is good enough  and  NP  if it isn t   It s helpful to know things about the problems we re trying to solve  rather than the algorithms we use to solve them  So we ll say that all the problems which have a well-scaling algorithm are  in P   And the ones which have a poor-scaling algorithm are  in NP    That means that lots of simple problems are  in NP  too  because we can write bad algorithms to solve easy problems  It would be good to know which problems in NP are the really tricky ones  but we don t just want to say  it s the ones we haven t found a good algorithm for   After all  I could come up with a problem  call it X  that I think needs a super-amazing algorithm  I tell the world that the best algorithm I could come up with to solve X scales badly  and so I think that X is a really tough problem  But tomorrow  maybe somebody cleverer than me invents an algorithm which solves X and is in P  So this isn t a very good definition of hard problems   All the same  there are lots of problems in NP that nobody knows a good algorithm for  So if I could prove that X is a certain sort of problem  one where a good algorithm to solve X could also be used  in some roundabout way  to give a good algorithm for every other problem in NP  Well now people might be a bit more convinced that X is a genuinely tricky problem  And in this case we call X NP-Complete

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It s a class of problems where we must simulate every possibility to be sure we have the optimal solution    There are a lot of good heuristics for some NP-Complete problems  but they are only an educated guess at best

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As far as I understand P is the set of problems which could be solved in polynomial time with a deterministic TM  NP is the set of problems which requires a non-deterministic TM to be solved in polynomial time  This means that we can check all different combinations of variables in parallel with each instance taking polynomial time  If the problem is solvable then at least one of those parallel instances of TM would halt with  quot yes quot   This also means that if you could make a correct guess about the variables solution then you just need to check it s validity in polynomial time  NP-Hard is the set where problems are at least as hard as NP  This means NP-Hard problem are equally or more difficult than any problem in NP set   Example of a NP-hard problem which is more difficult than NP is the halting problem  Example of a NP-hard problem which is equally difficult to a NP problem is any problem in NP-Complete set  see below    NP-Complete is the intersection set of NP and NP-Hard  A problems in NP-Complete set can be reduced to any other NP-Complete problem  That means if any of the NP-Complete problem would have an efficient solution then all of the NP-Complete problems could be solved with same solution  How can you prove P NP and win a US  1 000 000 prize  If you could show that every NP problem is reducible to any other NP problem  then NP-Complete set spans over entire NP sets  definition of NP-Complete   i e NP-Complete NP  Also since P NP  this means that at least one of those NP-Complete problem is solvable in polynomial time  which implies that entire NP set can be solved in polynomial time  Clearly it would conclude that P NP  Well this is just one way to prove P NP  there might be many other ways  Please let me know if I made any mistake

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NP-Complete means something very specific and you have to be careful or you will get the definition wrong   First  an NP problem is a yes no problem such that   There is polynomial-time proof for every instance of the problem with a  yes  answer that the answer is  yes   or  equivalently  There exists a polynomial-time algorithm  possibly using random variables  that has a non-zero probability of answering  yes  if the answer to an instance of the problem is  yes  and will say  no  100  of the time if the answer is  no   In other words  the algorithm must have a false-negative rate less than 100  and no false positives    A problem X is NP-Complete if   X is in NP  and For any problem Y in NP  there is a  reduction  from Y to X  a polynomial-time algorithm that transforms any instance of Y into an instance of X such that the answer to the Y-instance is  yes  if and only if the answer X-instance is  yes     If X is NP-complete and a deterministic  polynomial-time algorithm exists that can solve all instances of X correctly  0  false-positives  0  false-negatives   then any problem in NP can be solved in deterministic-polynomial-time  by reduction to X    So far  nobody has come up with such a deterministic polynomial-time algorithm  but nobody has proven one doesn t exist  there s a million bucks for anyone who can do either  the is the P   NP problem    That doesn t mean that you can t solve a particular instance of an NP-Complete  or NP-Hard  problem   It just means you can t have something that will work reliably on all instances of a problem the same way you could reliably sort a list of integers   You might very well be able to come up with an algorithm that will work very well on all practical instances of a NP-Hard problem

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It s a class of problems where we must simulate every possibility to be sure we have the optimal solution    There are a lot of good heuristics for some NP-Complete problems  but they are only an educated guess at best

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If you re looking for an example of an NP-complete problem then I suggest you take a look at 3-SAT   The basic premise is you have an expression in conjunctive normal form  which is a way of saying you have a series of expressions joined by ORs that all must be true    a or b  and  b or  c  and  d or  e or f        The 3-SAT problem is to find a solution that will satisfy the expression where each of the OR-expressions has exactly 3 booleans to match    a or  b or  c  and   a or b or  d  and  b or  c or d        A solution to this one might be  a T  b T  c F  d F   However  no algorithm has been discovered that will solve this problem in the general case in polynomial time  What this means is that the best way to solve this problem is to do essentially a brute force guess-and-check and try different combinations until you find one that works   What s special about the 3-SAT problem is that ANY NP-complete problem can be reduced to a 3-SAT problem  This means that if you can find a polynomial-time algorithm to solve this problem then you get  1 000 000  not to mention the respect and admiration of computer scientists and mathematicians around the world

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NP stands for Non-deterministic Polynomial time   This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine  like a regular Turing machine but also including a non-deterministic  choice  function   Basically  a solution has to be testable in poly time  If that s the case  and a known NP problem can be solved using the given problem with modified input  an NP problem can be reduced to the given problem  then the problem is NP complete   The main thing to take away from an NP-complete problem is that it cannot be solved in polynomial time in any known way  NP-Hard NP-Complete is a way of showing that certain classes of problems are not solvable in realistic time   Edit  As others have noted  there are often approximate solutions for NP-Complete problems  In this case  the approximate solution usually gives an approximation bound using special notation which tells us how close the approximation is

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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      Informally  an NP-complete problem is an NP problem that is at least as  tough  as any other problem in NP

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NP stands for Non-deterministic Polynomial time   This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine  like a regular Turing machine but also including a non-deterministic  choice  function   Basically  a solution has to be testable in poly time  If that s the case  and a known NP problem can be solved using the given problem with modified input  an NP problem can be reduced to the given problem  then the problem is NP complete   The main thing to take away from an NP-complete problem is that it cannot be solved in polynomial time in any known way  NP-Hard NP-Complete is a way of showing that certain classes of problems are not solvable in realistic time   Edit  As others have noted  there are often approximate solutions for NP-Complete problems  In this case  the approximate solution usually gives an approximation bound using special notation which tells us how close the approximation is

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Honestly  Wikipedia might be the best place to look for an answer to this   If NP   P  then we can solve very hard problems much faster than we thought we could before   If we solve only one NP-Complete problem in P  polynomial  time  then it can be applied to all other problems in the NP-Complete category

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NP Problem  -  NP problem are such problem that can be solved in non-deterministic polynomial time  Non deterministic algorithm operate in two stage  Non deterministic guessing stage  amp  amp  Non deterministic verification stage   Type of Np Problem  NP complete NP Hard  NP Complete problem  - 1 Decision Problem A is called NP complete if it has following two properties -  It belong to class NP  Every other problem in NP can be transformed to P in polynomial time   Some Ex  -  Knapsack problem sub set sum problem Vertex covering problem

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Basically this world s problems can be categorized as                   1  Unsolvable Problem                  2  Intractable Problem                  3  NP-Problem                  4  P-Problem                     1 The first one is no solution to the problem                   2 The second is the need exponential time  that is O  2   n  above                    3 The third is called the NP                   4 The fourth is easy problem     P  refers to a solution of the problem of Polynomial Time   NP  refers Polynomial Time yet to find a solution  We are not sure there is no Polynomial Time solution  but once you provide a solution  this solution can be verified in Polynomial Time   NP Complete  refers in Polynomial Time we still yet to find a solution  but it can be verified in Polynomial Time   The problem NPC in NP is the more difficult problem  so if we can prove that we have P solution to NPC problem then NP problems that can be found in P solution   NP Hard  refers Polynomial Time is yet to find a solution  but it sure is not able to be verified in Polynomial Time   NP Hard problem surpasses NPC difficulty

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It s a class of problems where we must simulate every possibility to be sure we have the optimal solution    There are a lot of good heuristics for some NP-Complete problems  but they are only an educated guess at best

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NP-Complete means something very specific and you have to be careful or you will get the definition wrong   First  an NP problem is a yes no problem such that   There is polynomial-time proof for every instance of the problem with a  yes  answer that the answer is  yes   or  equivalently  There exists a polynomial-time algorithm  possibly using random variables  that has a non-zero probability of answering  yes  if the answer to an instance of the problem is  yes  and will say  no  100  of the time if the answer is  no   In other words  the algorithm must have a false-negative rate less than 100  and no false positives    A problem X is NP-Complete if   X is in NP  and For any problem Y in NP  there is a  reduction  from Y to X  a polynomial-time algorithm that transforms any instance of Y into an instance of X such that the answer to the Y-instance is  yes  if and only if the answer X-instance is  yes     If X is NP-complete and a deterministic  polynomial-time algorithm exists that can solve all instances of X correctly  0  false-positives  0  false-negatives   then any problem in NP can be solved in deterministic-polynomial-time  by reduction to X    So far  nobody has come up with such a deterministic polynomial-time algorithm  but nobody has proven one doesn t exist  there s a million bucks for anyone who can do either  the is the P   NP problem    That doesn t mean that you can t solve a particular instance of an NP-Complete  or NP-Hard  problem   It just means you can t have something that will work reliably on all instances of a problem the same way you could reliably sort a list of integers   You might very well be able to come up with an algorithm that will work very well on all practical instances of a NP-Hard problem

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Basically this world s problems can be categorized as                   1  Unsolvable Problem                  2  Intractable Problem                  3  NP-Problem                  4  P-Problem                     1 The first one is no solution to the problem                   2 The second is the need exponential time  that is O  2   n  above                    3 The third is called the NP                   4 The fourth is easy problem     P  refers to a solution of the problem of Polynomial Time   NP  refers Polynomial Time yet to find a solution  We are not sure there is no Polynomial Time solution  but once you provide a solution  this solution can be verified in Polynomial Time   NP Complete  refers in Polynomial Time we still yet to find a solution  but it can be verified in Polynomial Time   The problem NPC in NP is the more difficult problem  so if we can prove that we have P solution to NPC problem then NP problems that can be found in P solution   NP Hard  refers Polynomial Time is yet to find a solution  but it sure is not able to be verified in Polynomial Time   NP Hard problem surpasses NPC difficulty

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The definitions for NP complete problems above is correct  but I thought I might wax lyrical about their philosophical importance as nobody has addressed that issue yet   Almost all complex problems you ll come up against will be NP Complete  There s something very fundamental about this class  and which just seems to be computationally different from easily solvable problems  They sort of have their own flavour  and it s not so hard to recognise them  This basically means that any moderately complex algorithm is impossible for you to solve exactly -- scheduling  optimising  packing  covering etc   But not all is lost if a problem you ll encounter is NP Complete  There is a vast and very technical field where people study approximation algorithms  which will give you guarantees for being close to the solution of an NP complete problem  Some of these are incredibly strong guarantees -- for example  for 3sat  you can get a 7 8 guarantee through a really obvious algorithm  Even better  in reality  there are some very strong heuristics  which excel at giving great answers  but no guarantees   for these problems   Note that two very famous problems -- graph isomorphism and factoring -- are not known to be P or NP

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We need to separate algorithms and problems  We write algorithms to solve problems  and they scale in a certain way  Although this is a simplification  let s label an algorithm with a  P  if the scaling is good enough  and  NP  if it isn t   It s helpful to know things about the problems we re trying to solve  rather than the algorithms we use to solve them  So we ll say that all the problems which have a well-scaling algorithm are  in P   And the ones which have a poor-scaling algorithm are  in NP    That means that lots of simple problems are  in NP  too  because we can write bad algorithms to solve easy problems  It would be good to know which problems in NP are the really tricky ones  but we don t just want to say  it s the ones we haven t found a good algorithm for   After all  I could come up with a problem  call it X  that I think needs a super-amazing algorithm  I tell the world that the best algorithm I could come up with to solve X scales badly  and so I think that X is a really tough problem  But tomorrow  maybe somebody cleverer than me invents an algorithm which solves X and is in P  So this isn t a very good definition of hard problems   All the same  there are lots of problems in NP that nobody knows a good algorithm for  So if I could prove that X is a certain sort of problem  one where a good algorithm to solve X could also be used  in some roundabout way  to give a good algorithm for every other problem in NP  Well now people might be a bit more convinced that X is a genuinely tricky problem  And in this case we call X NP-Complete

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What is NP  NP is the set of all decision problems  questions with a yes-or-no answer  for which the  yes -answers can be verified in polynomial time  O nk  where n is the problem size  and k is a constant  by a deterministic Turing machine  Polynomial time is sometimes used as the definition of fast or quickly  What is P  P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine  Since they can be solved in polynomial time  they can also be verified in polynomial time  Therefore P is a subset of NP  What is NP-Complete  A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly  ie  in polynomial time  transformed into x  In other words   x is in NP  and Every problem in NP is reducible to x  So  what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly  then all NP problems can be solved quickly  See also the post What s  quot P NP  quot   and why is it such a famous question  What is NP-Hard  NP-Hard are problems that are at least as hard as the hardest problems in NP  Note that NP-Complete problems are also NP-hard  However not all NP-hard problems are NP  or even a decision problem   despite having NP as a prefix  That is the NP in NP-hard does not mean non-deterministic polynomial time  Yes  this is confusing  but its usage is entrenched and unlikely to change

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Honestly  Wikipedia might be the best place to look for an answer to this   If NP   P  then we can solve very hard problems much faster than we thought we could before   If we solve only one NP-Complete problem in P  polynomial  time  then it can be applied to all other problems in the NP-Complete category

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