8,235 research outputs found
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
The N-K Problem in Power Grids: New Models, Formulations and Numerical Experiments (extended version)
Given a power grid modeled by a network together with equations describing
the power flows, power generation and consumption, and the laws of physics, the
so-called N-k problem asks whether there exists a set of k or fewer arcs whose
removal will cause the system to fail. The case where k is small is of
practical interest. We present theoretical and computational results involving
a mixed-integer model and a continuous nonlinear model related to this
question.Comment: 40 pages 3 figure
Using decision problems in public key cryptography
There are several public key establishment protocols as well as complete
public key cryptosystems based on allegedly hard problems from combinatorial
(semi)group theory known by now. Most of these problems are search problems,
i.e., they are of the following nature: given a property P and the information
that there are objects with the property P, find at least one particular object
with the property P. So far, no cryptographic protocol based on a search
problem in a non-commutative (semi)group has been recognized as secure enough
to be a viable alternative to established protocols (such as RSA) based on
commutative (semi)groups, although most of these protocols are more efficient
than RSA is.
In this paper, we suggest to use decision problems from combinatorial group
theory as the core of a public key establishment protocol or a public key
cryptosystem. By using a popular decision problem, the word problem, we design
a cryptosystem with the following features: (1) Bob transmits to Alice an
encrypted binary sequence which Alice decrypts correctly with probability "very
close" to 1; (2) the adversary, Eve, who is granted arbitrarily high (but
fixed) computational speed, cannot positively identify (at least, in theory),
by using a "brute force attack", the "1" or "0" bits in Bob's binary sequence.
In other words: no matter what computational speed we grant Eve at the outset,
there is no guarantee that her "brute force attack" program will give a
conclusive answer (or an answer which is correct with overwhelming probability)
about any bit in Bob's sequence.Comment: 12 page
Symmetry-breaking Answer Set Solving
In the context of Answer Set Programming, this paper investigates
symmetry-breaking to eliminate symmetric parts of the search space and,
thereby, simplify the solution process. We propose a reduction of disjunctive
logic programs to a coloured digraph such that permutational symmetries can be
constructed from graph automorphisms. Symmetries are then broken by introducing
symmetry-breaking constraints. For this purpose, we formulate a preprocessor
that integrates a graph automorphism system. Experiments demonstrate its
computational impact.Comment: Proceedings of ICLP'10 Workshop on Answer Set Programming and Other
Computing Paradig
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