Search results for "PACKING"
showing 10 items of 150 documents
Dichotomies properties on computational complexity of S-packing coloring problems
2015
This work establishes the complexity class of several instances of the S -packing coloring problem: for a graph G , a positive integer k and a nondecreasing list of integers S = ( s 1 , ? , s k ) , G is S -colorable if its vertices can be partitioned into sets S i , i = 1 , ? , k , where each S i is an s i -packing (a set of vertices at pairwise distance greater than s i ). In particular we prove a dichotomy between NP-complete problems and polynomial-time solvable problems for lists of at most four integers.
On Packing Colorings of Distance Graphs
2014
International audience; The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=…
Measure and dimension functions: measurability and densities
1997
During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whe…
A note on the packing of two copies of some trees into their third power
2003
Abstract It is proved in [1] that if a tree T of order n is not a star, then there exists an edge-disjoint placement of two copies of this tree into its fourth power. In this paper, we prove the packing of some trees into their third power.
An optimal extension of Marstrand?s plane-packing theorem
2003
We prove that if F is a subset of the 2-dimensional unit sphere in $\mathbb{R}^3$, with Hausdorff dimension strictly greater than 1, and E is a subset of $\mathbb{R}^3$ such that for each $e \in F$, E contains a plane perpendicular to the vector e, then E must have positive 3-dimensional Lebesgue measure.
Local dimensions of measures on infinitely generated self-affine sets
2014
We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space. We also give an estimate, that holds for all translation vectors, with only assuming the affine maps to be contractive.
A branch & bound algorithm for cutting and packing irregularly shaped pieces
2013
Abstract Cutting and packing problems involving irregular shapes, usually known as Nesting Problems, are common in industries ranging from clothing and footwear to furniture and shipbuilding. Research publications on these problems are relatively scarce compared with other cutting and packing problems with rectangular shapes, and are focused mostly on heuristic approaches. In this paper we make a systematic study of the problem and develop an exact Branch & Bound Algorithm. The initial existing mixed integer formulations are reviewed, tested and used as a starting point to develop a new and more efficient formulation. We also study several branching strategies, lower bounds and procedures f…
Electrophoresis of model colloidal spheres in low salt aqueous suspension
2004
We report on comprehensive measurements of the electrophoretic mobility μ of a highly charged spherical colloid in deionized or low salt aqueous suspensions, where fluid and crystalline order develops with increased packing fraction Φ. We propose the existence of a 'universal' shape of the μ(Φ) showing three distinct regimes: a logarithmic increase, a plateau and a logarithmic decrease. The position and the height of the plateau are found to be influenced by the particle surface properties and the electrolyte concentration. In particular, it starts once the counter-ion concentration becomes equal to the concentration of background electrolyte. This coincides only loosely with the range of Φ…
RutaRep: a computer package to design dispatching routes in the meat industry
2005
In this paper we present a computer program that has been developed to design the dispatching routes of a medium-sized meat company in Spain. We have modelled the real problem as a variant of the vehicle routing problem with Time Windows and implemented a number of heuristic algorithms based on the most advanced solution techniques for this problem. These algorithms have been embedded in a computer package that is intended to be used as a decision support system for the distribution manager. The program runs under Windows System and is straightforward to use. We also present some computational experiences based on real instances provided by the company. This experience shows important impro…
A GRASP algorithm for the container stowage slot planning problem
2016
This work presents a generalization of the Slot Planning Problem which raises when the liner shipping industry needs to plan the placement of containers within a vessel (stowage planning). State-of-the-art stowage planning relies on a heuristic decomposition where containers are first distributed in clusters along the vessel. For each of those clusters a specific position for each container must be found. Compared to previous studies, we have introduced two new features: the explicit handling of rolled out containers and the inclusion of separations rules for dangerous cargo. We present a novel integer programming formulation and a Greedy Randomized Adaptive Search Procedure (GRASP) to solv…