Greg Cohen | Searcharxiv

arXiv2017

Some facts on Permanents in Finite Characteristics

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.

cs.CC↗

arXiv2010

A new algebraic technique for polynomial-time computing the number modulo 2 of Hamiltonian decompositions and similar partitions of a graph's edge set

In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the number modulo 2 of, say, Hamiltonian cycles/paths etc. While the problems of finding a Hamiltonian decomposition and Hamiltonian cycle are NP-complete, counting these objects modulo 2 in polynomial time is yet possible for certain types of regular undirected graphs. Some of the most known examples are the theorems about the existence of an even number of Hamiltonian decompositions in a 4-regular graph and an even number of such decompositions where two given edges e and g belong to different cycles (Thomason, 1978), as well as an even number of Hamiltonian cycles passing through any given edge in a regular odd-degreed graph (Smith's theorem). The present article introduces a new algebraic technique which generalizes the notion of counting modulo 2 via applying fields of Characteristic 2 and determinants and, for instance, allows to receive a polynomial-time formula for the number modulo 2 of a 4-regular bipartite graph's Hamiltonian decompositions such that a given edge and a given path of length 2 belong to different Hamiltonian cycles - hence refining/extending (in a computational sense) Thomason's result for bipartite graphs. This technique also provides a polynomial-time calculation of the number modulo 2 of a graph's edge set decompositions into simple cycles each containing at least one element of a given set of its edges what is a similar kind of extension of Thomason's theorem as well.

cs.DM↗