A Spinorial Formulation of the Maximum Clique Problem of a Graph

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors, that is vectors of the endomorphism space of Clifford algebras, and the maximum clique problem is formalized in this setting so that, this much studied problem, may take advantage from recent progresses of pure spinor geometry

On Computational Complexity of Clifford Algebra

After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced in a previous paper.Comment: 13 page

Nouveaux algorithmes, bornes et formulations pour les problèmes de la clique maximum et de la coloration minimum

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Variable Formulation and Neighborhood Search Methods for the Maximum Clique Problem in Graph

Doktorska disertacija se bavi temama rešavanja računarski teških problema kombinatorne optimizacije. Istaknut je problem maksimalne klike kao predstavnik određenih struktura u grafovima. Problem maksimalne klike i sa njim povezani problemi su formulisani kao nelinearne funkcije. Rešavani su sa ciljem otkrivanja novih metoda koje pronalaze dobre aproksimacije rešenja za neko razumno vreme. Predložene su varijante Metode promenljivih okolina na rešavanje maksimalne klike u grafu. Povezani problemi na grafovima se mogu primeniti na pretragu informacija, raspoređivanje, procesiranje signala, teoriju klasifikacije, teoriju kodiranja, itd. Svi algoritmi su implementirani i uspešno testirani na brojnim različitim primerima.This Ph.D. thesis addresses topics NP hard problem solving approaches in combinatorial optimization and according to that it is highlighted maximum clique problem as a representative of certain structures in graphs. Maximum clique problem and related problems with this have been formulated as non linear functions which have been solved to research for new methods and good solution approximations for some reasonable time. It has been proposed several different extensions of Variable Neighborhood Search method. Related problems on graphs could be applied on information retrieval, scheduling, signal processing, theory of classi_cation, theory of coding, etc. Algorithms are implemented and successfully tested on various different tasks

Mathematical and physical aspects of complex symmetric operators

Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables

Glosarium Matematika

273 p.; 24 cm