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
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
space cannot use more than
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.
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. Descriptive Complexity, , Neil, Immerman, Springer-Verlag, 1999, 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. Descriptive Complexity, , Neil, Immerman, Springer-Verlag, 1999, ISBN 0-387-98600-6
6. Complexity of Relational Query Languages, , M., Vardi, 14th Symposium on Theory of Computation, 1983
7. Relational Queries Computable in Polynomial Time, , Neil, Immerman, 14th ACM STOC Symp., 1982 . Revised version in ''Information and Control'', 68 (1986), 86-104.
8. Theory of Computation, , Dexter C., Kozen, Springer, 2006, 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.
★
Introduction to the Theory of Computation, Michael Sipser, , , PWS Publishing, 1997, ISBN 0-534-94728-X Section 7.2: The Class P, pp.234–241.
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