Computational problems | Linear algebra | Computational complexity theory | Matrix theory | Permutations

Computing the permanent

In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions. 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, it is generally believed that the permanent cannot be computed in polynomial time. 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 . 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. The development of both exact and approximate algorithms for computing the permanent of a matrix is an active area of research. (Wikipedia).

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Related pages

European Journal of Combinatorics | Permanent (mathematics) | Random self-reducibility | Randomized algorithm | ACC0 | Linear algebra | Perfect matching | Pfaffian orientation | Planar graph | Hamiltonian path | FKT algorithm | Boson sampling | Tutte matrix | Permutation | Markov chain Monte Carlo | Metropolis–Hastings algorithm | Symmetric group | Homeomorphism (graph theory) | Determinant | Minor (linear algebra) | Gaussian elimination | Discrete uniform distribution | Polarization identity | Complete bipartite graph | Parity of a permutation | Bipartite graph | Inclusion–exclusion principle | Markov chain | NP (complexity) | Markov chain mixing time | Unitary matrix | Approximation algorithm | George Pólya | Gray code | Pfaffian | P versus NP problem | Sampling (statistics) | Journal of Combinatorial Theory | Symmetric tensor | Computational complexity theory | Square root | Skew-symmetric matrix | Matrix (mathematics) | Modular arithmetic | Invertible matrix | UP (complexity)