Complexity classes P and NP

The relation between the complexity classes P and NP is one of the most important open problems in theoretical computer science and mathematics. It is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes to solve a problem).

In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP consists of all those decision problems whose positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine. Arguably, the biggest open question in theoretical computer science concerns the relationship between those two classes:

Is P equal to NP?

In a 2002 poll of 100 researchers, 61 believed the answer is no, 9 believed the answer is yes, 22 were unsure, and 8 believed the question may be independent of the currently accepted axioms, and so impossible to prove or disprove.^* The Clay Mathematics Institute has offered a USD 1,000,000 prize for a correct solution.

An important role in this discussion is played by the set of NP-complete problems (or NPC) which can be loosely described as the hardest problems in NP and therefore they are the least likely to be in P. More precisely, any problem in NP, through some efficient (takes at most a polynomial-bounded number of steps) transformation, can be expressed as a problem in NP-complete. Therefore if one finds an efficient (again, polynomial-bounded) solution to any NP-complete problem, then every problem in NP can be solved efficiently and therefore must be in P, hence proving P = NP. (See NP-complete for the exact definition.) Most theoretical computer scientists currently believe that the relationship among the classes P, NP, and NPC is as shown in the picture, with the P and NPC classes disjoint.

In essence, the P = NP question asks: if positive solutions to a YES/NO problem can be verified quickly, can the answers also be computed quickly? Here is an example to get a feeling for the question. Given a set of integers, does any subset of them sum to 0? For instance, does a subset of the set {-2, -3, 8, 15, -10} add up to 0? The answer is YES, though it may take a little while to find a subset that does - and if the set was larger, it might take a very long time to find a subset that does. On the other hand, if someone claims that the answer is "YES, because {-2, -3, -10, 15} add up to zero", then we can quickly check that with a few additions. Verifying that the subset adds up to zero is much faster than finding the subset in the first place. The information needed to verify a positive answer is also called a certificate. So we conclude that given the right certificates, positive answers to our problem can be verified quickly (i.e. in polynomial time) and that's why this problem is in NP.

The restriction to YES/NO problems doesn't really make a difference; even if we allow more complicated answers, the resulting problem (whether FP = FNP) is equivalent.

Formal definitions

More precisely, a decision problem is a problem that takes as input some string and requires as output either YES or NO. If there is an algorithm (say a Turing machine, or a Lisp or Pascal program with unbounded memory) which is able to produce the correct answer for any input string of length n in at most c*n^k steps, where k and c are some constants independent of the input string, then we say that the problem can be solved in polynomial time and we place it in the class P. Intuitively, we think of the problems in P as those that can be solved reasonably quickly.

Now suppose there is an algorithm A(w,C) which takes two arguments, a string w which is an input string to our decision problem, and a string C which is a "proposed certificate", and such that A produces a YES/NO answer in at most c*n^k steps (where n is the length of w and neither c nor k depend on w). Suppose furthermore that

w is a YES instance of the decision problem if and only if there exists C such that A(w,C) returns YES.

Then we say that the problem can be solved in non-deterministic polynomial time and we place it in the class NP. We think of the algorithm A as a verifier of proposed certificates which runs reasonably fast. (Note that the abbreviation NP stands for "Non-deterministic Polynomial" and not for "Non-Polynomial".)

NP-complete

To attack the P = NP question, the concept of NP-completeness is very useful. Informally, the NP-complete problems are the "toughest" problems in NP in the sense that they are the ones most likely not to be in P. NP-hard problems are those into which any problem in NP can be transformed, in polynomial time. NP-complete problems are those NP-hard problems which are in NP. For instance, the decision problem version of the traveling salesman problem is NP-complete. So any instance of any problem in NP can be transformed mechanically into an instance of the traveling salesman problem, in polynomial time. So, if the traveling salesman problem turned out to be in P, then P = NP! The traveling salesman problem is one of many such NP-complete problems. If any NP-complete problem is in P, then it would follow that P = NP. Unfortunately, many important problems have been shown to be NP-complete and not a single fast algorithm for any of them is known.

Still harder problems

Although it is unknown whether P = NP, problems outside of P are known. A number of succinct problems, that is, problems which operate not on normal input but on a computational description of the input, are known to be EXPTIME-complete. Because it can be shown that P ⊂ EXPSPACE, these problems are outside P, and so require more than polynomial time. In fact, by the time hierarchy theorem, they cannot be solved in significantly less than exponential time.

The problem of deciding the truth of a statement in Presburger arithmetic is even harder. Fischer and Rabin proved in 1974 that every algorithm which decides the truth of Presburger statements has a runtime of at least $2^{2^{cn}}$ for some constant c. Here, n is the length of the Presburger statement. Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the halting problem. They cannot be solved in general given any finite amount of time.

Is P really tractable?

All of the above discussion has assumed that P means "easy" and "not in P" means "hard". While this is a common and reasonably accurate assumption in complexity theory, it is not always true in practice, for several reasons:

It ignores constant factors. A problem that takes time 10¹⁰⁰⁰n is in P (it is linear time), but is completely intractable in practice. A problem that takes time 10^-100002ⁿ is not in P (it is exponential time), but is very tractable for values of n up into the thousands.
It ignores the size of the exponents. A problem with time n¹⁰⁰⁰ is in P, yet intractable. Problems have been proven to exist in P that require arbitrarily large exponents (see time hierarchy theorem). A problem with time 2^n/1000 is not in P, yet is tractable for n up into the thousands.
It only considers worst-case times. There might be a problem that arises in the real world such that most of the time, it can be solved in time n, but on very rare occasions you'll see an instance of the problem that takes time 2ⁿ. This problem might have an average time that is polynomial, but the worst case is exponential, so the problem wouldn't be in P.
It only considers deterministic solutions. Imagine a problem that you can solve quickly if you accept a tiny error probability, but a guaranteed correct answer is much harder to get. The problem would not belong to P even though in practice it can be solved quickly. This is in fact a common approach to attack problems in NP not known to be in P (see RP, BPP). Even if P=BPP, as many researchers believe, it is often considerably easier to find probabilistic algorithms.
New computing models such as quantum computers may be able to quickly solve some problems not known to be in P; however, none of the problems they are known to be able to solve are NP-hard. However, the definition of P and NP are in terms of classical computing models like Turing machines. Therefore, even if a quantum computer algorithm were discovered to efficiently solve an NP-hard problem, we would only have a way of physically solving difficult problems quickly, not a proof that the mathematical classes P and NP are equal.

Why do computer scientists think P ≠ NP?

Most computer scientists believe that P≠NP. A key reason for this belief is that after decades of studying these problems, nobody has been able to find a polynomial-time algorithm for an NP-hard problem. Moreover, these algorithms were sought long before the concept of NP-completeness was even known (Karp's 21 NP-complete problems, among the first found, were all well-known existing problems). Furthermore, the result P = NP would imply many other startling results that are currently believed to be false, such as NP = co-NP and P = PH.

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.

On the other hand, some researchers believe that we are overconfident in P ≠ NP and should explore proofs of P = NP as well. For example, in 2002 these statements were made: ^*

The main argument in favour of P≠NP is the total lack of fundamental progress in the area of exhaustive search. 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. . . The resolution of Fermat's Last Theorem also shows that very simply ^* questions may be settled only by very deep theories.

— Moshe Vardi, Rice 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.

— Anil Nerode, Cornell University

Consequences of proof

One of the reasons why the problem attracts so much attention are the consequences of the answer.

A proof of P = NP could have stunning practical consequences, if the proof leads to efficient methods for solving every problem in NP. Many NP-complete problems are fundamental in many fields. One negative impact is that many forms of cryptography, notably public-key cryptography, would become trivially easy to "crack" and would therefore be useless. However, there are also 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, to name two of the most famous examples. Efficient solutions to these problems would have enormous implications for logistics. But it is by no means the only field that would be profoundly changed. To take one of very many examples, important problems in Protein structure prediction are NP-complete ^*; if these problems were solvable efficiently it could spur considerable advances in biology.

But such changes may pale into insigificance compared to the revolution an efficient method for solving NP-complete problems would cause in mathematics itself. According to Stephen Cook ^* (PDF),

...it would transform mathematics by allowing a computer to find a formal proof of any theorem which has a proof of a reasonable length, since formal proofs can easily be recognized in polynomial time. 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 and transorm mathematics into a search for useful things to prove.

A proof that showed that P ≠ NP would clearly not have such stunning implications, but would represent a massive advance in computational complexity theory.

Results about difficulty of proof

A million-dollar prize and a huge amount of dedicated research with no substantial results are enough to show the problem is difficult. There have also been some formal results demonstrating why the problem might be difficult to solve.

One of the most frequently-cited is a result involving oracles. Imagine you have a magical machine called an oracle that can solve only one problem, such as determining if a given number is prime, but can solve it in constant time. Our new question is now, if we're allowed to use this oracle as much as we want, are there problems we can verify in polynomial time that we can't solve in polynomial time? It turns out that, depending on the problem that the oracle solves, with certain oracles one has P = NP, while for other oracles one has P ≠ NP. The practical consequence of this is that any proof which can be modified to account for the existence of these oracles cannot solve the problem. Unfortunately, most known methods and nearly all classical methods can be modified in such a way (we say they are relativizing).

Furthermore, a 1993 result by Alexander Razborov and Steven Rudich showed that, given a certain credible assumption, proofs that are "natural" in a certain sense cannot solve the P = NP problem (see natural proof). This demonstrated that some of the most seemingly-promising methods of the time were also unlikely to succeed. As more theorems of this kind are proved, a potential proof of the theorem has more and more traps to avoid.

This is actually another reason why NP-complete problems are useful: if a polynomial-time algorithm, or the lack of one, can be demonstrated for an NP-complete problem, this would solve the P = NP problem in a way which is not excluded by the above results.

Polynomial-time algorithms

No one knows whether polynomial-time algorithms exist for NP-complete languages. But if such algorithms do exist, we already know some of them! For example, the following algorithm correctly accepts an NP-complete language, but no one knows how long it takes in general. This is a polynomial-time algorithm if and only if P = NP.

// Algorithm that accepts the NP-complete language SUBSET-SUM. // // This is a polynomial-time algorithm if and only if P=NP. // // "Polynomial-time" means it returns "YES" in polynomial time when // the answer should be "YES", and runs forever when it's "NO". // // Input: S = a finite set of integers // Output: "YES" if any subset of S adds up to 0. // Otherwise, it runs forever with no output. // Note: "Program number P" is the program you get by // writing the integer P in binary, then // considering that string of bits to be a // program. Every possible program can be // generated this way, though most do nothing // because of syntax errors.
FOR N = 1...infinity FOR P = 1...N Run program number P for N steps with input S IF the program outputs a list of distinct integers AND the integers are all in S AND the integers sum to 0
THEN OUTPUT "YES" and HALT

If P = NP, then this is a polynomial-time algorithm accepting an NP-Complete language. "Accepting" means it gives "YES" answers in polynomial time, but is allowed to run forever when the answer is "NO".

Perhaps we want to "solve" the SUBSET-SUM problem, rather than just "accept" the SUBSET-SUM language. That means we want it to always halt and return a "YES" or "NO" answer. Does any algorithm exist that can provably do this in polynomial time? No one knows. But if such algorithms do exist, then we already know some of them! Just replace the IF statement in the above algorithm with this:

IF the program outputs a complete math proof AND each step of the proof is legal AND the conclusion is that S does (or does not) have a subset summing to 0 THEN OUTPUT "YES" (or "NO" if that was proved) and HALT

Logical characterizations

The P=NP problem can be restated in terms of the expressibility of certain classes of logical statements. All languages in P can be expressed in first-order logic with the addition of a least fixed point operator (effectively, this allows the definition of recursive functions). Similarly, NP is the set of languages expressible in existential second-order logic — that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets. The languages in the polynomial hierarchy, PH, correspond to all of second-order logic. Thus, the question "is P a proper subset of NP" can be reformulated as "is existential second-order logic able to describe languages that first-order logic with least fixed point cannot?"

Trivia

The Princeton University computer science building has the question "P=NP?" encoded in binary in its brickwork on the top floor of the west side. If it is proven that P=NP, the bricks can easily be changed to encode "P=NP!". If P does not equal NP, it can be changed to "P<NP!". ^*

Hubert Chen, PhD, of Cornell University offers this tongue-in-cheek proof that P does not equal NP: "Proof by contradiction. Assume P = NP. Let y be a proof that P = NP. The proof y can be verified in polynomial time by a competent computer scientist, the existence of which we assert. However, since P = NP, the proof y can be generated in polynomial time by such computer scientists. Since this generation has not yet occurred (despite attempts by such computer scientists to produce a proof), we have a contradiction."^*

The P = NP problem has also been featured in television:

In a scene of The Simpsons entitled "Homer³" (part of the Treehouse of Horror VI episode), Homer enters the third dimension where "P = NP" appears as a hovering equation in this bizarre parallel universe.

In an episode of Futurama, Fry and Amy spend a moment in a supply closet, in which there are two separate binders labelled "P" and "NP".

In the second episode of the CBS show NUMB3RS, Charlie, a mathematician, works with his brother, an FBI agent, to predict which bank a group of seemingly non-violent robbers will hit next. When FBI's attempt to arrest the criminals ends in bloodshed, Charlie tries to deal with his emotional reaction by attempting to solve P = NP. (The show used the popular computer game Minesweeper to help explain what he was working on.)

Ryan North's Dinosaur Comics strip #43 deals with a "P=NP?" lecture.

References

A. S. Fraenkel and D. Lichtenstein, Computing a perfect strategy for n*n chess requires time exponential in n, Proc. 8th Int. Coll. Automata, Languages, and Programming, Springer LNCS 115 (1981) 278-293 and J. Comb. Th. A 31 (1981) 199-214.
E. Berlekamp and D. Wolfe, Mathematical Go: Chilling Gets the Last Point, A. K. Peters, 1994. D. Wolfe, Go endgames are hard, MSRI Combinatorial Game Theory Research Worksh., 2000.
Neil Immerman. Languages Which Capture Complexity Classes. 15th ACM STOC Symposium, pp.347-354. 1983.

External links

The Clay Math Institute Millennium Prize Problems
The Clay Math Institute Official Problem Description (pdf)
Gerhard J. Woeginger. The P-versus-NP page. A list of links to a number of purported solutions to the problem. Some of these links state that P equals NP, some of them state the opposite. It's probable that all these alleged solutions are incorrect.
Computational Complexity of Games and Puzzles
The "PRIMES is in P" FAQ
Scott Aaronson's Complexity Zoo: P, NP
Qeden, a wiki that aims to solve the Millennium Prize Problems

Complexity classes | Conjectures | Unsolved problems in mathematics | Unsolved problems in computer science | Millennium Prize Problems

Gourt Home::Gourt :: Complexity classes P and NP