PPA (complexity) explained

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).

Definition

PPA is defined as follows. Suppose we have a graph on whose vertices are

n -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]

Notes and References

1. Christos Papadimitriou . 1994 . On the complexity of the parity argument and other inefficient proofs of existence . . 48 . 3 . 498–532 . 10.1016/S0022-0000(05)80063-7 . 2009-12-19 . https://web.archive.org/web/20160304084618/http://www.cs.berkeley.edu/~christos/papers/On%20the%20Complexity.pdf . 2016-03-04 . dead .
2. Michelangelo Grigni . 1995 . A Sperner Lemma Complete for PPA . . 77 . 5–6 . 255–259 . 10.1016/S0020-0190(00)00152-6. 10.1.1.63.9463 .
3. A. Filos-Ratsikas . P.W. Goldberg . Consensus-Halving is PPA-Complete . . 2018 . 51–64 . 10.1145/3188745.3188880 . 1711.04503 .
4. E. Jeřábek . Integer Factoring and Modular Square Roots . . 2016 . 82 . 2 . 380–394 . 10.1016/j.jcss.2015.08.001 . 1207.5220 .