31,084 research outputs found
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers
In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
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