52,319 research outputs found
Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression
We present an exact method, based on an arc-flow formulation with side
constraints, for solving bin packing and cutting stock problems --- including
multi-constraint variants --- by simply representing all the patterns in a very
compact graph. Our method includes a graph compression algorithm that usually
reduces the size of the underlying graph substantially without weakening the
model. As opposed to our method, which provides strong models, conventional
models are usually highly symmetric and provide very weak lower bounds.
Our formulation is equivalent to Gilmore and Gomory's, thus providing a very
strong linear relaxation. However, instead of using column-generation in an
iterative process, the method constructs a graph, where paths from the source
to the target node represent every valid packing pattern.
The same method, without any problem-specific parameterization, was used to
solve a large variety of instances from several different cutting and packing
problems. In this paper, we deal with vector packing, graph coloring, bin
packing, cutting stock, cardinality constrained bin packing, cutting stock with
cutting knife limitation, cutting stock with binary patterns, bin packing with
conflicts, and cutting stock with binary patterns and forbidden pairs. We
report computational results obtained with many benchmark test data sets, all
of them showing a large advantage of this formulation with respect to the
traditional ones
Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming
We have formulated the problem of generating periodic dense paritcle packings
as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation
[S. Torquato and Y. Jiao, Phys. Rev. E {\bf 80}, 041104 (2009)]. Because the
objective function and impenetrability constraints can be exactly linearized
for sphere packings with a size distribution in -dimensional Euclidean space
, it is most suitable and natural to solve the corresponding ASC
optimization problem using sequential linear programming (SLP) techniques. We
implement an SLP solution to produce robustly a wide spectrum of jammed sphere
packings in for and with a diversity of disorder
and densities up to the maximally densities. This deterministic algorithm can
produce a broad range of inherent structures besides the usual disordered ones
with very small computational cost by tuning the radius of the {\it influence
sphere}. In three dimensions, we show that it can produce with high probability
a variety of strictly jammed packings with a packing density anywhere in the
wide range . We also apply the algorithm to generate various
disordered packings as well as the maximally dense packings for
and 6. Compared to the LS procedure, our SLP protocol is able to ensure that
the final packings are truly jammed, produces disordered jammed packings with
anomalously low densities, and is appreciably more robust and computationally
faster at generating maximally dense packings, especially as the space
dimension increases.Comment: 34 pages, 6 figure
Defragmenting the Module Layout of a Partially Reconfigurable Device
Modern generations of field-programmable gate arrays (FPGAs) allow for
partial reconfiguration. In an online context, where the sequence of modules to
be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of
modules leads to progressive fragmentation of the available space, making
defragmentation an important issue. We address this problem by propose an
online and an offline component for the defragmentation of the available space.
We consider defragmenting the module layout on a reconfigurable device. This
corresponds to solving a two-dimensional strip packing problem. Problems of
this type are NP-hard in the strong sense, and previous algorithmic results are
rather limited. Based on a graph-theoretic characterization of feasible
packings, we develop a method that can solve two-dimensional defragmentation
instances of practical size to optimality. Our approach is validated for a set
of benchmark instances.Comment: 10 pages, 11 figures, 1 table, Latex, to appear in "Engineering of
Reconfigurable Systems and Algorithms" as a "Distinguished Paper
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