421 research outputs found
The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect
In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances.Two-dimensional cutting, defect, AND/OR-graph, Branch & Bound
Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube
We study a two-parameter generalization of the Catalan numbers:
is the number of ways to subdivide the -dimensional hypercube into
rectangular blocks using orthogonal partitions of fixed arity . Bremner \&
Dotsenko introduced in their work on Boardman--Vogt tensor
products of operads; they used homological algebra to prove a recursive formula
and a functional equation. We express as simple finite sums, and
determine their growth rate and asymptotic behaviour. We give an elementary
proof of the functional equation, using a bijection between hypercube
decompositions and a family of full -ary trees. Our results generalize the
well-known correspondence between Catalan numbers and full binary trees
On the number of rectangulations of a planar point set
AbstractWe investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n+1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schröder number, and the total number of rectangulations is O(20n/n4)
The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect
In this paper, a two-dimensional cutting problem is considered in which a single plate (large object) has to be cut down into a set of small items of maximal value. As opposed to standard cutting problems, the large object contains a defect, which must not be covered by a small item. The problem is represented by means of an AND/OR-graph, and a Branch & Bound procedure (including heuristic modifications for speeding up the search process) is introduced for its exact solution. The proposed method is evaluated in a series of numerical experiments that are run on problem instances taken from the literature, as well as on randomly generated instances
Permutation of elements in double semigroups
Double semigroups have two associative operations related by
the interchange relation: . Kock \cite{Kock2007} (2007) discovered a
commutativity property in degree 16 for double semigroups: associativity and
the interchange relation combine to produce permutations of elements. We show
that such properties can be expressed in terms of cycles in directed graphs
with edges labelled by permutations. We use computer algebra to show that 9 is
the lowest degree for which commutativity occurs, and we give self-contained
proofs of the commutativity properties in degree 9.Comment: 24 pages, 11 figures, 4 tables. Final version accepted by Semigroup
Forum on 12 March 201
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