P¹

In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.

P is often taken to be the class of computational problems which are "efficiently solvable" or "tractable", although there are potentially larger classes that are also considered tractable such as RP and BPP. Also, there exist problems in P which are intractable in practical terms; for example, some require at least n^1000000 operations. See even harder problems of complexity classes for further discussion.

Notable problems in P

P is known to contain many natural problems, including the decision versions of linear programming, calculating the greatest common divisor, and finding a maximum matching. In 2002, it was shown that the problem of determining if a number is prime is in P.^[1] The related class of function problems is FP.

Several natural problems are complete for P, including st-connectivity (or reachability) on alternating graphs.^[2] The article on P-complete problems lists further relevant problems in P.

Relationships to other classes

A generalization of P is NP, which is the class of languages decidable in polynomial time on a non-deterministic Turing machine. We then trivially have P is a subset of NP. Though unproven, most experts believe this is a strict subset.^[3]

P is also known to be at least as large as L, the class of problems decidable in a logarithmic amount of memory space. A decider using O(logn) space cannot use more than 2^O(logn) = n^O(1) time, because this is the total number of possible configurations; thus, L is a subset of P. Another important problem is whether L = P. We do know that P = AL, the set of problems solvable in logarithmic memory by alternating Turing machines. P is also known to be no larger than PSPACE, the class of problems decidable in polynomial space. Again, whether P = PSPACE is an open problem. To summarize:

Here, EXPTIME is the class of problems solvable in exponential time. Of all the classes shown above, only two strict containments are known:

• P is strictly contained in EXPTIME. Consequently, all EXPTIME-hard problems lie outside P, and at least one of the containments to the right of P above is strict (in fact, it is widely believed that all three are strict).
• L is strictly contained in PSPACE.

The most difficult problems in P are P-complete problems.

Another generalization of P is P/poly, or Nonuniform Polynomial-Time. If a problem is in P/poly, then it can be solved in deterministic polynomial time provided that an advice string is given that depends only on the length of the input. Unlike for NP, however, the polynomial-time machine doesn't need to detect fraudulent advice strings; it is not a verifier. P/poly is a large class containing nearly all practical algorithms, including all of BPP. If it contains NP, then the polynomial hierarchy collapses to the second level. On the other hand, it also contains some impractical algorithms, including some undecidable problems such as the unary version of any undecidable problem.

In 1999, Jin-Yi Cai and D. Sivakumar, building on work by Ogihara, showed that if there exists a sparse language which is P-complete, then L = P.^[4]

Properties

Polynomial-time algorithms are closed under composition. Intuitively, this says that if I write a function which is polynomial-time assuming that function calls are constant-time, and if those called functions themselves require polynomial time, then the entire algorithm takes polynomial time. One consequence of this is that P is low for itself. This is also one of the main reasons that P is considered to be a machine-independent class; any machine "feature", such as random access, which can be simulated in polynomial time can simply be composed with the main polynomial-time algorithm to reduce it to a polynomial-time algorithm on a more basic machine.

Pure existence proofs of polynomial-time algorithms

Some problems are known to be solvable in polynomial-time, but no concrete algorithm is known for solving them. For example, the Robertson-Seymour theorem guarantees that there is a finite list of forbidden minors that characterizes (for example) the set of graphs that can be embedded on a torus; moreover, Robertson and Seymour showed that there is an O(n³) algorithm for determinining whether a graph has a given graph as a minor. This yields a nonconstructive proof that there is a polynomial-time algorithm for determining if a given graph can be embedded on a torus, despite the fact that no concrete algorithm is known for this problem.

Alternative characterizations

In descriptive complexity, P can be described as the problems expressible in FO (LFP), the class of first-order logic with a least fixed point operator added to it. In Immerman's 1999 textbook on descriptive complexity^[5], Immerman ascribes this result to Vardi 1982^[6] and to Immerman 1982^[7].

History

Kozen^[8] states that Cobham and Edmonds are "generally credited with the invention of the notion of polynomial time".

References

1. ^ Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P", Annals of Mathematics 160 (2004), no. 2, pp. 781–793.
2. ^ Immerman, Neil (1999). Descriptive Complexity. New York: Springer-Verlag. ISBN 0-387-98600-6. 
3. ^ Johnsonbaugh, Richard; Schaefer, Marcus, Algorithms, 2004 Pearson Education, page 458, ISBN 0-02-360692-4
4. ^ Jin-Yi Cai and D. Sivakumar. Sparse hard sets for P: resolution of a conjecture of Hartmanis. Journal of Computer and System Sciences, volume 58, issue 2, pp.280–296. 1999. ISSN:0022-0000. At Citeseer
5. ^ Immerman, Neil (1999). Descriptive Complexity. New York: Springer-Verlag, 66. ISBN 0-387-98600-6. 
6. ^ Vardi, M. (1983). "Complexity of Relational Query Languages". 14th Symposium on Theory of Computation: 137-146. 
7. ^ Immerman, Neil (1982). "Relational Queries Computable in Polynomial Time". 14th ACM STOC Symp.: 147-152. . Revised version in Information and Control, 68 (1986), 86-104.
8. ^ Kozen, Dexter C. (2006). Theory of Computation. Springer, 4. ISBN 1-84628-297-7. 

• C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994. ISBN 0-201-53082-1.
• Complexity Zoo: P, P/poly
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 34.1: Polynomial time, pp.971–979.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X.  Section 7.2: The Class P, pp.234–241.

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