Enumerations of specific permutation classes: Difference between revisions

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'''PPA''' is a [[complexity class]], standing for "Polynomial Parity Argument" (on a graph). Introduced by [[Christos Papadimitriou]] in 1994<ref>{{cite journal | author = Christos Papadimitriou | year = 1994 | title = On the complexity of the parity argument and other inefficient proofs of existence | journal = [[Journal of Computer and System Sciences]] | volume = 48 | issue = 3 | pages = 498–532 | url = http://www.cs.berkeley.edu/~christos/papers/On%20the%20Complexity.pdf | doi = 10.1016/S0022-0000(05)80063-7}}</ref> (page 528), PPA is a subclass of [[TFNP]]. It is a class of search problems that can be shown to be total by an application of the [[handshaking lemma]]: ''any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number''. This observation means that if we are given a graph and an odd-degree vertex, and we are asked to find some other odd-degree vertex, then we are searching for something that is guaranteed to exist (so, we have a total search problem).


PPA is defined as follows. Suppose we have a graph on whose vertices are <math>n</math>-bit binary strings, and the graph is represented by a polynomial-sized circuit that takes a vertex as input and outputs its neighbors. (Note that this allows us to represent an exponentially-large graph on which we can efficiently perform local exploration.) Suppose furthermore that a specific vertex (say the all-zeroes vector) has an odd number of neighbors. We are required to find another odd-degree vertex. Note that this problem is in NP—given a solution it may be verified using the circuit that the solution is correct.  A function computation problem belongs to PPA if it admits a [[polynomial-time reduction]] to this graph search problem. A problem is [[Complete (complexity)|complete]] for the class PPA if in addition, this graph search problem is reducible to that problem.


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PPA contains [[PPAD]] as a subclass. This is because the corresponding problem that defines PPAD, known as END OF THE LINE, can be reduced (in a straightforward way) to the above search for an additional odd-degree vertex (essentially, just by ignoring the directions of the edges in END OF THE LINE).
 
There is an un-oriented version of the Sperner lemma known to be complete for PPA.<ref>{{cite journal | author = Michelangelo Grigni | year = 1995 | title = A Sperner Lemma Complete for PPA | journal = [[Information Processing Letters]] | volume = 77 | pages = 255–259 | doi = 10.1016/S0020-0190(00)00152-6}}</ref> The problem of searching for a second [[Hamiltonian cycle]] on a 3-regular graph is a member of PPA, but is not known to be complete for PPA.
 
==References==
{{reflist}}
 
[[Category:Complexity classes]]

Revision as of 14:42, 1 November 2013

PPA is a complexity class, standing for "Polynomial Parity Argument" (on a graph). Introduced by Christos Papadimitriou in 1994[1] (page 528), PPA is a subclass of TFNP. It is a class of search problems that can be shown to be total by an application of the handshaking lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number. This observation means that if we are given a graph and an odd-degree vertex, and we are asked to find some other odd-degree vertex, then we are searching for something that is guaranteed to exist (so, we have a total search problem).

PPA is defined as follows. Suppose we have a graph on whose vertices are -bit binary strings, and the graph is represented by a polynomial-sized circuit that takes a vertex as input and outputs its neighbors. (Note that this allows us to represent an exponentially-large graph on which we can efficiently perform local exploration.) Suppose furthermore that a specific vertex (say the all-zeroes vector) has an odd number of neighbors. We are required to find another odd-degree vertex. Note that this problem is in NP—given a solution it may be verified using the circuit that the solution is correct. A function computation problem belongs to PPA if it admits a polynomial-time reduction to this graph search problem. A problem is complete for the class PPA if in addition, this graph search problem is reducible to that problem.

PPA contains PPAD as a subclass. This is because the corresponding problem that defines PPAD, known as END OF THE LINE, can be reduced (in a straightforward way) to the above search for an additional odd-degree vertex (essentially, just by ignoring the directions of the edges in END OF THE LINE).

There is an un-oriented version of the Sperner lemma known to be complete for PPA.[2] The problem of searching for a second Hamiltonian cycle on a 3-regular graph is a member of PPA, but is not known to be complete for PPA.

References

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