PPA (complexity)

In computational complexity theory, 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).

## Contents

* 1 Definition
* 2 Related classes
* 3 Examples
* 4 References

## Definition

PPA is defined as follows. Suppose we have a graph on whose vertices are [math]\displaystyle{ 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 for the class PPA if in addition, this graph search problem is reducible to that problem.

## Related classes

PPAD is defined in a similar way to PPA, except that it is defined on directed graphs. PPAD is a subclass of PPA. 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).

## Examples

* There is an un-oriented version of the Sperner lemma known to be complete for PPA.[2]
* The consensus-halving problem is known to be complete for PPA.[3]
* 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.
* There is a randomized polynomial-time reduction from the problem of integer factorization to problems complete for PPA.[4]

## References

1. ↑ Christos Papadimitriou (1994). "On the complexity of the parity argument and other inefficient proofs of existence". Journal of Computer and System Sciences 48 (3): 498–532. doi:10.1016/S0022-0000(05)80063-7. http://www.cs.berkeley.edu/~christos/papers/On the Complexity.pdf. Retrieved 2009-12-19.
2. ↑ Michelangelo Grigni (1995). "A Sperner Lemma Complete for PPA". Information Processing Letters 77 (5–6): 255–259. doi:10.1016/S0020-0190(00)00152-6.
3. ↑ A. Filos-Ratsikas; P.W. Goldberg (2018). "Consensus-Halving is PPA-Complete". pp. 51–64. doi:10.1145/3188745.3188880.
4. ↑ E. Jeřábek (2016). "Integer Factoring and Modular Square Roots". Journal of Computer and System Sciences 82 (2): 380–394. doi:10.1016/j.jcss.2015.08.001.

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