HOME

TheInfoList



OR:

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, the computation of the permanent of a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
is a problem that is thought to be more difficult than the computation of the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
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 In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...
by
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...
, it is generally believed that the permanent cannot be computed in polynomial time. In
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
, 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.


Definition and naive algorithm

The permanent of an ''n''-by-''n'' matrix ''A'' = (''a''''i,j'') is defined as : \operatorname(A)=\sum_\prod_^n a_. The sum here extends over all elements σ of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
''S''''n'', i.e. over all
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s 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 . Ryser's method is based on an inclusion–exclusion formula that can be given as follows: Let A_k be obtained from ''A'' by deleting ''k'' columns, let P(A_k) be the product of the row-sums of A_k, and let \Sigma_k be the sum of the values of P(A_k) over all possible A_k. Then : \operatorname(A) = \sum_^ (-1)^k \Sigma_k. It may be rewritten in terms of the matrix entries as follows : \operatorname (A) = (-1)^n \sum_ (-1)^ \prod_^n \sum_ a_. Ryser's formula can be evaluated using O(2^n^2) arithmetic operations, or O(2^n) by processing the sets S in
Gray code The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray (researcher), Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For ...
order.


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 , ; also . 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 In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product t ...
for a
symmetric tensor In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tens ...
. 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 . The simplest known formula of this type (when the characteristic of the field is not two) is : \operatorname(A) = \frac \left sum_\delta \left(\prod_^n \delta_k\right) \prod_^n \sum_^n \delta_i a_\right where the outer sum is over all 2^ vectors \delta=(\delta_1=1,\delta_2,\dots,\delta_n)\in \^n.


Special cases


Planar and ''K''3,3-free

The number of
perfect matching In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph with edges and vertices , a perfect matching in is a subset of , such that every vertex in is adjacent to exact ...
s in a
bipartite graph In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...
is counted by the permanent of the graph's
biadjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...
, 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 graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
s (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 In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a ...
(the
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of its
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
) is the number of perfect matchings. This technique can be generalized to graphs that contain no subgraph
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the
complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...
''K''3,3.
George Pólya George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...
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 () that there is no linear map T such that \operatorname T(A) = \det A for all n\times n matrices A. The characterization of "convertible" matrices was given by who showed that such matrices are precisely those that are the biadjacency matrix of bipartite graphs that have a Pfaffian orientation: an orientation of the edges such that for every even cycle C for which G\setminus C has a perfect matching, there are an odd number of edges directed along C (and thus an odd number with the opposite orientation). It was also shown that these graphs are exactly those that do not contain a subgraph homeomorphic to K_, as above.


Computation modulo a number

Modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
2, the permanent is the same as the determinant, as (-1) \equiv 1 \pmod 2. It can also be computed modulo 2^k in time O(n^) for k \ge 2. However, it is UP-hard to compute the permanent modulo any number that is not a power of 2. There are various formulae given by for the computation modulo a prime . First, there is one using symbolic calculations with partial derivatives. Second, for = 3 there is the following formula for an n×n-matrix A, involving the matrix's principal minors (): :\operatorname (A) = (-1)^n \sum_ \det(A_J)\det(A_\bar), where A_J is the submatrix of A induced by the rows and columns of A indexed by J, and \bar J is the complement of J in \, while the determinant of the empty submatrix is defined to be 1. The expansion above can be generalized in an arbitrary characteristic ''p'' as the following pair of dual identities: \begin \operatorname (A) &= (-1)^n \sum_ \det(A_)\dotsm\det(A_) \\ \det (A) &= (-1)^n \sum_ \operatorname(A_)\dotsm\operatorname(A_) \end where in both formulas the sum is taken over all the (''p'' − 1)-tuples , \ldots, that are partitions of the set \ into ''p'' − 1 subsets, some of them possibly empty. The former formula possesses an analog for the hafnian of a symmetric A and an odd p: \operatorname^2(A) = (-1)^n \sum_ \det(A_)\dotsm\det(A_)(-1)^ with the sum taken over the same set of indexes. Moreover, in characteristic zero a similar convolution sum expression involving both the permanent and the determinant yields the
Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
polynomial (defined as \operatorname(A) = \sum_\prod_^n a_ where H_n is the set of n-permutations having only one cycle): \operatorname (A) = \sum_ \det(A_J)\operatorname(A_\bar)(-1)^. In characteristic 2 the latter equality turns into \operatorname (A) = \sum_ \det(A_J)\operatorname(A_\bar) what therefore provides an opportunity to polynomial-time calculate the
Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
polynomial of any
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...
U (i.e. such that U^\textsf U = I where I is the identity ''n''×''n''-matrix), because each minor of such a matrix coincides with its algebraic complement: \operatorname (U) = \operatorname^2(U + I_) where I_ is the identity ''n''×''n''-matrix with the entry of indexes 1,1 replaced by 0. Moreover, it may, in turn, be further generalized for a unitary ''n''×''n''-matrix U as \operatorname (U) = \operatorname^2(U + I_) where K is a subset of , I_ is the identity ''n''×''n''-matrix with the entries of indexes ''k'',''k'' replaced by 0 for all ''k'' belonging to K, and we define \operatorname(A) = \sum_ \prod_^n a_ where H_n(K) is the set of n-permutations whose each cycle contains at least one element of K. This formula also implies the following identities over fields of characteristic 3: for any
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
A :\operatorname(A^)\operatorname^2(A) = \operatorname(A); for any
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...
U, that is, a square matrix U such that U^\textsf U = I where I is the identity matrix of the corresponding size, :\operatorname^2(U) = \det(U + V)\det(-U) where V is the matrix whose entries are the cubes of the corresponding entries of U. It was also shown () that, if we define a square matrix A as k-semi-unitary when \operatorname(A^\textsf A - I) = k, the permanent of a 1-semi-unitary matrix is computable in polynomial time over fields of characteristic 3, while for > 1 the problem becomes #3-P-complete. (A parallel theory concerns the
Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
polynomial in characteristic 2: while computing it on the unitary matrices is polynomial-time feasible, the problem is #2-P-complete for the k-semi-unitary ones for any ''k'' > 0). The latter result was essentially extended in 2017 () and it was proven that in characteristic 3 there is a simple formula relating the permanents of a square matrix and its partial inverse (for A_ and A_ being square, A_ being
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
): \operatorname\beginA_ & A_\\ A_ & A_\end = \operatorname^2(A_)\operatorname\beginA_^ & A_^A_\\ A_A_^ & A_-A_A_^A_ \end and it allows to polynomial-time reduce the computation of the permanent of an ''n''×''n''-matrix with a subset of ''k'' or ''k'' − 1 rows expressible as linear combinations of another (disjoint) subset of k rows to the computation of the permanent of an (''n'' − ''k'')×(''n'' − ''k'')- or (''n'' − ''k'' + 1)×(''n'' − ''k'' + 1)-matrix correspondingly, hence having introduced a compression operator (analogical to the Gaussian modification applied for calculating the determinant) that "preserves" the permanent in characteristic 3. (Analogically, it would be worth noting that the
Hamiltonian cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
polynomial in characteristic 2 does possess its invariant matrix compressions as well, taking into account the fact that ham(''A'') = 0 for any ''n''×''n''-matrix ''A'' having three equal rows or, if ''n'' > 2, a pair of indexes ''i'',''j'' such that its ''i''-th and ''j''-th rows are identical and its ''i''-th and ''j''-th columns are identical too.) The closure of that operator defined as the limit of its sequential application together with the transpose transformation (utilized each time the operator leaves the matrix intact) is also an operator mapping, when applied to classes of matrices, one class to another. While the compression operator maps the class of 1-semi-unitary matrices to itself and the classes of
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...
and 2-semi-unitary ones, the compression-closure of the 1-semi-unitary class (as well as the class of matrices received from unitary ones through replacing one row by an arbitrary row vector — the permanent of such a matrix is, via the Laplace expansion, the sum of the permanents of 1-semi-unitary matrices and, accordingly, polynomial-time computable) is yet unknown and tensely related to the general problem of the permanent's computational complexity in characteristic 3 and the chief question of
P versus NP The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, "quickly" means an algorithm exists that ...
: as it was shown in (), if such a compression-closure is the set of all square matrices over a field of characteristic 3 or, at least, contains a matrix class the permanent's computation on is #3-P-complete (like the class of 2-semi-unitary matrices) then the permanent is computable in polynomial time in this characteristic. Besides, the problem of finding and classifying any possible analogs of the permanent-preserving compressions existing in characteristic 3 for other prime characteristics was formulated (), while giving the following identity for an ''n''×''n'' matrix A and two ''n''-vectors (having all their entries from the set ) \alpha and \beta such that , valid in an arbitrary prime characteristic ''p'': \operatorname(A^) = \det^(A) \operatorname(A^)^ \left(\prod_^n\alpha_i!\right) \left(\prod_^n\beta_j!\right) (-1)^ where for an ''n''×''m''-matrix M, an n-vector x and an m-vector y, both vectors having all their entries from the set , M^ denotes the matrix received from M via repeating x_i times its ''i''-th row for ''i'' = 1, ..., ''n'' and y_j times its ''j''-th column for ''j'' = 1, ..., ''m'' (if some row's or column's multiplicity equals zero it would mean that the row or column was removed, and thus this notion is a generalization of the notion of submatrix), and \vec_n denotes the n-vector all whose entries equal unity. This identity is an exact analog of the classical formula expressing a matrix's minor through a minor of its inverse and hence demonstrates (once more) a kind of duality between the determinant and the permanent as relative immanants. (Actually its own analogue for the hafnian of a symmetric A and an odd prime p is \operatorname^2(A^) = \det^(A) \operatorname^2(A^)^ \left(\prod_^n \alpha_i!\right)^2 (-1)^). And, as an even wider generalization for the partial inverse case in a prime characteristic p, for A_, A_ being square, A_ being
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
and of size x, and , there holds also the identity \operatorname \beginA_ & A_\\ A_ & A_\end ^ = ^(A_) \operatorname \begin A_^ & A_^ A_ \\ A_ A_^ & A_ - A_ A_^ A_\end^ \left(\prod_^n\alpha_!\right)\left(\prod_^n\beta_!\right) (-1)^ where the common row/column multiplicity vectors \alpha and \beta for the matrix A generate the corresponding row/column multiplicity vectors \alpha_s and \beta_t, s,t = 1,2, for its blocks (the same concerns A's partial inverse in the equality's right side).


Approximate computation

When the entries of ''A'' are nonnegative, the permanent can be computed approximately in
probabilistic Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
polynomial time, up to an error of ε''M'', 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) (). 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 In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that ...
algorithm that uses a Metropolis rule to define and run a
Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
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 . Let M(G) denote the number of perfect matchings in G. Roughly, for any particular edge e in G, by sampling many matchings in G and counting how many of them are matchings in G \setminus e, one can obtain an estimate of the ratio \rho=\frac. The number M(G) is then \rho M(G \setminus e), where M(G \setminus e) can be approximated by applying the same method recursively. Another class of matrices for which the permanent is of particular interest, is the positive-semidefinite matrices. Using a technique of Stockmeyer counting, they can be computed within the class \textsf^\textsf, but this is considered an infeasible class in general. It is NP-hard to approximate permanents of PSD matrices within a subexponential factor, and it is conjectured to be \textsf^\textsf-hard If further constraints on the
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
are imposed, there are more efficient algorithms known. One randomized algorithm is based on the model of boson sampling and it uses the tools proper to
quantum optics Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction ...
, to represent the permanent of positive-semidefinite matrices as the expected value of a specific random variable. The latter is then approximated by its sample mean. This algorithm, for a certain set of positive-semidefinite matrices, approximates their permanent in polynomial time up to an additive error, which is more reliable than that of the standard classical polynomial-time algorithm by Gurvits.


Notes


References

* * * * * * * * * * * * * * * * * * * * * *


Further reading

* . {{DEFAULTSORT:Computing The Permanent Computational complexity theory Linear algebra Matrix theory Permutations Computational problems