4,006 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
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
Approximating Bin Packing within O(log OPT * log log OPT) bins
For bin packing, the input consists of n items with sizes s_1,...,s_n in
[0,1] which have to be assigned to a minimum number of bins of size 1. The
seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT
+ O(log^2 OPT) bins.
We provide the first improvement in now 3 decades and show that one can find
a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is
achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation
using the Entropy Method from discrepancy theory. The result is constructive
via algorithms of Bansal and Lovett-Meka
Using Differential Evolution for the Graph Coloring
Differential evolution was developed for reliable and versatile function
optimization. It has also become interesting for other domains because of its
ease to use. In this paper, we posed the question of whether differential
evolution can also be used by solving of the combinatorial optimization
problems, and in particular, for the graph coloring problem. Therefore, a
hybrid self-adaptive differential evolution algorithm for graph coloring was
proposed that is comparable with the best heuristics for graph coloring today,
i.e. Tabucol of Hertz and de Werra and the hybrid evolutionary algorithm of
Galinier and Hao. We have focused on the graph 3-coloring. Therefore, the
evolutionary algorithm with method SAW of Eiben et al., which achieved
excellent results for this kind of graphs, was also incorporated into this
study. The extensive experiments show that the differential evolution could
become a competitive tool for the solving of graph coloring problem in the
future
Online Bin Packing with Advice
We consider the online bin packing problem under the advice complexity model
where the 'online constraint' is relaxed and an algorithm receives partial
information about the future requests. We provide tight upper and lower bounds
for the amount of advice an algorithm needs to achieve an optimal packing. We
also introduce an algorithm that, when provided with log n + o(log n) bits of
advice, achieves a competitive ratio of 3/2 for the general problem. This
algorithm is simple and is expected to find real-world applications. We
introduce another algorithm that receives 2n + o(n) bits of advice and achieves
a competitive ratio of 4/3 + {\epsilon}. Finally, we provide a lower bound
argument that implies that advice of linear size is required for an algorithm
to achieve a competitive ratio better than 9/8.Comment: 19 pages, 1 figure (2 subfigures
Probabilistic analysis of Online Bin Coloring algorithms via Stochastic Comparison
This paper proposes a new method for probabilistic analysis of online algorithms that is based on the notion of stochastic dominance. We develop the method for the Online Bin Coloring problem introduced by Krumke et al. Using methods for the stochastic comparison of Markov chains we establish the strong result that the performance of the online algorithm GreedyFit is stochastically dominated by the performance of the algorithm OneBin for any number of items processed. This result gives a more realistic picture than competitive analysis and explains the behavior observed in simulations.mathematical applications;
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