1,835 research outputs found
Enumeration of Circuits and Minimal Forbidden Sets
In resource-constrained scheduling, it is sometimes important to know all inclusion-minimal subsets of jobs that must not be scheduled simultaneously. These so-called minimal forbidden sets are given implicitly by a linear inequality system, and can be interpreted more generally as the circuits of a particular independence system. We present several complexity results related to computation, enumeration, and counting of the circuits of an independence system. On this account, we also propose a backtracking algorithm that enumerates all minimal forbidden sets for resource constrained scheduling problems
On the sum of the Voronoi polytope of a lattice with a zonotope
A parallelotope is a polytope that admits a facet-to-facet tiling of
space by translation copies of along a lattice. The Voronoi cell
of a lattice is an example of a parallelotope. A parallelotope can be
uniquely decomposed as the Minkowski sum of a zone closed parallelotope and
a zonotope , where is the set of vectors used to generate the
zonotope. In this paper we consider the related question: When is the Minkowski
sum of a general parallelotope and a zonotope a parallelotope? We give
two necessary conditions and show that the vectors have to be free. Given a
set of free vectors, we give several methods for checking if is
a parallelotope. Using this we classify such zonotopes for some highly
symmetric lattices.
In the case of the root lattice , it is possible to give a more
geometric description of the admissible sets of vectors . We found that the
set of admissible vectors, called free vectors, is described by the well-known
configuration of lines in a cubic. Based on a detailed study of the
geometry of , we give a simple characterization of the
configurations of vectors such that is a
parallelotope. The enumeration yields maximal families of vectors, which
are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table
An algorithm for counting circuits: application to real-world and random graphs
We introduce an algorithm which estimates the number of circuits in a graph
as a function of their length. This approach provides analytical results for
the typical entropy of circuits in sparse random graphs. When applied to
real-world networks, it allows to estimate exponentially large numbers of
circuits in polynomial time. We illustrate the method by studying a graph of
the Internet structure.Comment: 7 pages, 3 figures, minor corrections, accepted versio
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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