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In linear algebra, the permanent of an n-by-n matrix A=(a_i,j) is defined as

The sum here extends over all elements σ of the symmetric group S_n, i.e. over all permutations of the number 1,2,...,n.

For example,

The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account. If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). A formula similar to Laplace's for the development of a determinant along a row or column is also valid for the permanent; all signs have to be ignored for the permanent.

Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics. The permanent describes the number of perfect matchings in a bipartite graph. More specifically, let G be a bipartite graph with vertices A₁, A₂, ..., A_n on one side and B₁, B₂, ..., B_n on the other side. Then, G can be described by an n-by-n matrix A=(a_i,j) where a_i,j = 1 if there is an edge between the vertices A_i and B_j and a_i,j = 0 otherwise. The permanent of this matrix is equal to the number of perfect matchings in the graph.

The permanent is also more difficult to compute than the determinant. The determinant can be computed in polynomial time by Gaussian elimination. The permanent cannot be computed by Gaussian elimination. Moreover, computing the permanent of a 0-1 matrix (matrix whose entries are 0 or 1) is #P-complete. Thus, if the permanent can be computed in polynomial time by any method, then P= #PP pronounced "sharp P", is a complexity class in computational complexity theory. It is the set of counting problems associated with the decision problems in the set NP . Unlike most well-known complexity classes, it is not a class of decision problems bu which is an even stronger statement than PComputational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much m= NPNP may stand for: The complexity class NP in computational complexity theory; see NP (complexity) NP Buchverlag Noun phrase in grammar National park National Post, a Toronto-based newspaper Nepal ( ISO 3166-1 alpha-2 country code) Neptunium ( chemical sym. It can, however, be computed approximatelyIn computer science, approximation algorithms are an approach to attacking NP-hard optimization problems. Since it is unlikely that there can ever be efficient exact algorithms solving NP-hard problems, one settles for non-optimal solutions, but requires in probabilisticAlgorithms A randomized algorithm is an algorithm which is allowed to flip a truly random coin. In common practice, this means that the machine implementing the algorithm has access to a pseudo-random number generator. The algorithm typically uses the ran polynomial time, up to an error of εM, where M is the value of the permanent and ε>0 is arbitrary.

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