Some facts on Permanents in Finite Characteristics

📝 Abstract

The polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that the differences of their Gram matrices and the corresponding identity matrices are of rank k) in the case k = 0 or k = 1 and its #3P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely the case k > 1 regarding the (n-k)x(n-k)-sub-permanents (or permanent-minors) of a unitary nxn-matrix and their possible relations, because an (n-k)x(n-k)-submatrix of a unitary nxn-matrix is generically a k-semi-unitary (n-k)x(n-k)-matrix. The following paper offers a way to receive a variety of such equations of different sorts, in the meantime extending this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for the sub-permanents of a generic matrix they might yield, including a number of generalizations and formulae (valid in an arbitrary prime characteristic) analogical to the classical identities relating the minors of a matrix and its inverse. Moreover, the second chapter also deals with the Hamiltonian cycle polynomial in characteristic 2 that surprisingly possesses quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3, while over the field GF(2) it obtains even more amazing features. Besides, the third chapter is devoted to the computational complexity issues of the permanent and some related functions on a variety of Cauchy matrices and their certain generalizations, including constructing a polynomial-time algorithm (based on them) for the permanent of an arbitrary matrix in characteristic 5 (implying RP = NP) and conjecturing the existence of a similar scheme in characteristic 3.

💡 Analysis

The polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that the differences of their Gram matrices and the corresponding identity matrices are of rank k) in the case k = 0 or k = 1 and its #3P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely the case k > 1 regarding the (n-k)x(n-k)-sub-permanents (or permanent-minors) of a unitary nxn-matrix and their possible relations, because an (n-k)x(n-k)-submatrix of a unitary nxn-matrix is generically a k-semi-unitary (n-k)x(n-k)-matrix. The following paper offers a way to receive a variety of such equations of different sorts, in the meantime extending this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for the sub-permanents of a generic matrix they might yield, including a number of generalizations and formulae (valid in an arbitrary prime characteristic) analogical to the classical identities relating the minors of a matrix and its inverse. Moreover, the second chapter also deals with the Hamiltonian cycle polynomial in characteristic 2 that surprisingly possesses quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3, while over the field GF(2) it obtains even more amazing features. Besides, the third chapter is devoted to the computational complexity issues of the permanent and some related functions on a variety of Cauchy matrices and their certain generalizations, including constructing a polynomial-time algorithm (based on them) for the permanent of an arbitrary matrix in characteristic 5 (implying RP = NP) and conjecturing the existence of a similar scheme in characteristic 3.

📄 Content

Anna Knezevic Greg Cohen Marina Domanskaya

Some Facts on Permanents in Finite Characteristics

Abstract:
The permanent’s polynomial-time computability over fields of characteristic 3 for k-semi- unitary matrices (i.e. n×n-matrices A such that 𝑟𝑎𝑛𝑘(𝐴𝐴𝑇−𝐼𝑛) = 𝑘) in the case k ≤ 1 and its #3P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely the case k > 1 regarding the (n-k)×(n-k)- sub-permanents (or permanent-minors) of a unitary n×n-matrix and their possible relations, because an (n-k)×(n-k)-submatrix of a unitary n×n-matrix is generically a k- semi-unitary (n-k)×(n-k)-matrix. The following paper offers a way to receive a variety of such equations of different sorts, in the meantime extending (in its second chapter divided into subchapters) this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for a generic matrix’s sub-permanents they might yield, including a number of generalizations and formulae (valid in an arbitrary prime characteristic) analogical to the classical identities relating the minors of a matrix and its inverse. Moreover, the second chapter also deals with the Hamiltonian cycle polynomial in characteristic 2 that surprisingly demonstrates quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3, while in the field GF(2) it obtains even more amazing features that are extensions of many well-known results on the parity of Hamiltonian cycles. Besides, the paper’s third chapter is devoted to the computational complexity issues of the permanent and some related functions on a variety of Cauchy matrices and their certain generalizations, including constructing a polynomial-time algorithm (based on them) for the permanent of an arbitrary square matrix in characteristic 5 and conjecturing the existence of a similar scheme in characteristic 3. Throughout the paper, we investigate various matrix compressions and transformations preserving the permanent and related functions in certain finite characteristics. And, as an auxiliary algebraic tool supposed for an application when needed in all the constructions we’re going to discuss in the present article, we’ll introduce and utilize a special principle involving a field’s extension by a formal infinitesimal and allowing, provided a number of conditions are fulfilled, to reduce the computation of a polynomial over a field to solving a system of algebraic equations in polynomial time.

Introduction Historically the computation of polynomials over finite fields was considered as quiet a special area related to the general theory of computational complexity. It’s known that the existence of a polynomial-time algorithm for computing the number of solutions of an NP-complete problem modulo p (i.e. the statement that the complexity class #pP is a subset of P) implies the equality RP = NP for any prime p. This fact can be demonstrated via considering, for instance, the Hamiltonian cycle polynomial ham(Z) over a finite field F, where Z is an n×n-matrix, that is a homogeneous polynomial in Z’s entries such that each variable’s degree is 0 or 1 in each monomial. In the meantime, any polynomial over F in m variables such that each variable’s degree is 0 or 1 in each monomial can have no more than m|F|m−1 roots over F (it’s easy to prove by the induction on m). Hence if we take an n×n-matrix W = {wi,j}n×n over F and, given a digraph G with n vertices whose adjacency matrix is AG, define its weighted adjacency matrix as AG ⋆W, where ⋆ denotes the Hadamard (entry-wise) product, then the equation for the variables wi,j ham(AG ⋆W) = 0 can have no more than n2|F|n2−1 roots over |F|. It implies that if we consider W random then the probability that ham(AG ⋆W) = 0 is smaller than n2 |F| when G is Hamiltonian and 1 otherwise. Moreover, because for all the 2n2 possible adjacency matrices AG we can get, altogether, no more than 2n2n2|F|n2−1 roots of the equations ham(AG ⋆W) = 0, in case if 2n2n2 |F| < 1 it also implies the existence of a matrix W such that for any digraph G with n vertices ham(AG ⋆W) = 0 if and only if G isn’t Hamiltonian, while the probability of taking such a matrix W randomly is 1 − 2nn2 |F| . Accordingly it also demonstrates the known fact that RP is a subset of P/poly. On the other hand, given a finite field F of characteristic p, any computational circuit can be polynomial-time represented as a set of relations over F where each variable is either expressed as the sum or product of two other variables or equaled to a given constant. Such a representation hence calculates a polynomial in the set of given constants. In the meantime, when we extend F to a bigger field F̂ we therefore receive an extension of this p

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