In linear algebra, the computation of the permanent of a matrix is a problem that is known to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions.
Contents
- Definition and naive algorithm
- Ryser formula
- Balasubramanian Bax Franklin Glynn formula
- Planar and K33 free
- Computation modulo a number
- Approximate computation
- References
The permanent is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns. However, where the determinant weights each of these products with a ±1 sign based on the parity of the set, the permanent weights them all with a +1 sign.
While the determinant can be computed in polynomial time by Gaussian elimination, the permanent cannot. In computational complexity theory, a theorem of Valiant states that computing permanents is #P-hard, and even #P-complete for matrices in which all entries are 0 or 1.Valiant (1979) This puts the computation of the permanent in a class of problems believed to be even more difficult to compute than NP. It is known that computing the permanent is impossible for logspace-uniform ACC0 circuits.(Allender & Gore 1994)
The development of both exact and approximate algorithms for computing the permanent of a matrix is an active area of research.
Definition and naive algorithm
The permanent of an n-by-n matrix A = (ai,j) is defined as
The sum here extends over all elements σ of the symmetric group Sn, i.e. over all permutations of the numbers 1, 2, ..., n. This formula differs from the corresponding formula for the determinant only in that, in the determinant, each product is multiplied by the sign of the permutation σ while in this formula each product is unsigned. The formula may be directly translated into an algorithm that naively expands the formula, summing over all permutations and within the sum multiplying out each matrix entry. This requires n! n arithmetic operations.
Ryser formula
The best known general exact algorithm is due to H. J. Ryser (1963). Ryser’s method is based on an inclusion–exclusion formula that can be given as follows: Let
It may be rewritten in terms of the matrix entries as follows
Ryser’s formula can be evaluated using
Balasubramanian-Bax-Franklin-Glynn formula
Another formula that appears to be as fast as Ryser's (or perhaps even twice as fast) is to be found in the two Ph.D. theses; see (Balasubramanian 1980), (Bax 1998); also (Bax & Franklin 1996). The methods to find the formula are quite different, being related to the combinatorics of the Muir algebra, and to finite difference theory respectively. Another way, connected with invariant theory is via the polarization identity for a symmetric tensor (Glynn 2010). The formula generalizes to infinitely many others, as found by all these authors, although it is not clear if they are any faster than the basic one. See (Glynn 2013).
The simplest known formula of this type (when the characteristic of the field is not two) is
where the outer sum is over all
Planar and K3,3-free
The number of perfect matchings in a bipartite graph is counted by the permanent of the graph's biadjacency matrix, and the permanent of any 0-1 matrix can be interpreted in this way as the number of perfect matchings in a graph. For planar graphs (regardless of bipartiteness), the FKT algorithm computes the number of perfect matchings in polynomial time by changing the signs of a carefully chosen subset of the entries in the Tutte matrix of the graph, so that the Pfaffian of the resulting skew-symmetric matrix (the square root of its determinant) is the number of perfect matchings. This technique can be generalized to graphs that contain no subgraph homeomorphic to the complete bipartite graph K3,3.
George Pólya had asked the question of when it is possible to change the signs of some of the entries of a 01 matrix A so that the determinant of the new matrix is the permanent of A. Not all 01 matrices are "convertible" in this manner; in fact it is known (Marcus & Minc (1961)) that there is no linear map
Computation modulo a number
Modulo 2, the permanent is the same as the determinant, as
There are various formulae given by Glynn (2010) for the computation modulo a prime
Secondly, for
where
(The determinant of an empty submatrix is defined to be 1).
This formula implies the following identities over fields of Characteristic 3 (Grigoriy Kogan, 1996):
for any invertible
for any unitary
where
Approximate computation
When the entries of A are nonnegative, the permanent can be computed approximately in probabilistic polynomial time, up to an error of εM, where M is the value of the permanent and ε > 0 is arbitrary. In other words, there exists a fully polynomial-time randomized approximation scheme (FPRAS) (Jerrum, Vigoda & Sinclair (2001)).
The most difficult step in the computation is the construction of an algorithm to sample almost uniformly from the set of all perfect matchings in a given bipartite graph: in other words, a fully polynomial almost uniform sampler (FPAUS). This can be done using a Markov chain Monte Carlo algorithm that uses a Metropolis rule to define and run a Markov chain whose distribution is close to uniform, and whose mixing time is polynomial.
It is possible to approximately count the number of perfect matchings in a graph via the self-reducibility of the permanent, by using the FPAUS in combination with a well-known reduction from sampling to counting due to Jerrum, Valiant & Vazirani (1986). Let