78 research outputs found
Locality-preserving allocations Problems and coloured Bin Packing
We study the following problem, introduced by Chung et al. in 2006. We are
given, online or offline, a set of coloured items of different sizes, and wish
to pack them into bins of equal size so that we use few bins in total (at most
times optimal), and that the items of each colour span few bins (at
most times optimal). We call such allocations -approximate. As usual in bin packing problems, we allow additive
constants and consider as the asymptotic performance ratios.
We prove that for \eps>0, if we desire small , no scheme can beat
(1+\eps, \Omega(1/\eps))-approximate allocations and similarly as we desire
small , no scheme can beat (1.69103, 1+\eps)-approximate allocations.
We give offline schemes that come very close to achieving these lower bounds.
For the online case, we prove that no scheme can even achieve
-approximate allocations. However, a small restriction on item
sizes permits a simple online scheme that computes (2+\eps, 1.7)-approximate
allocations
Online Bin Covering: Expectations vs. Guarantees
Bin covering is a dual version of classic bin packing. Thus, the goal is to
cover as many bins as possible, where covering a bin means packing items of
total size at least one in the bin.
For online bin covering, competitive analysis fails to distinguish between
most algorithms of interest; all "reasonable" algorithms have a competitive
ratio of 1/2. Thus, in order to get a better understanding of the combinatorial
difficulties in solving this problem, we turn to other performance measures,
namely relative worst order, random order, and max/max analysis, as well as
analyzing input with restricted or uniformly distributed item sizes. In this
way, our study also supplements the ongoing systematic studies of the relative
strengths of various performance measures.
Two classic algorithms for online bin packing that have natural dual versions
are Harmonic and Next-Fit. Even though the algorithms are quite different in
nature, the dual versions are not separated by competitive analysis. We make
the case that when guarantees are needed, even under restricted input
sequences, dual Harmonic is preferable. In addition, we establish quite robust
theoretical results showing that if items come from a uniform distribution or
even if just the ordering of items is uniformly random, then dual Next-Fit is
the right choice.Comment: IMADA-preprint-c
Improved space for bounded-space, on-line bin-packing
The author presents a sequence of linear-time, bounded-space, on-line, bin-packing algorithms that are based on the "HARMONIC" algorithms {\text{H}}_k introduced by Lee and Lee [J. Assoc. Comput. Mach., 32 (1985), pp. 562–572]. The algorithms in this paper guarantee the worst case performance of {\text{H}}_k, whereas they only use $O ( \log \log k ) instead of k active bins. For k\geqq 6, the algorithms in this paper outperform all known heuristics using k active bins. For example, the author gives an algorithm that has worst case ratio less than 17/10 and uses only six active bin
Sparse, continuous policy representations for uniform online bin packing via regression of interpolants
Online bin packing is a classic optimisation problem, widely tackled by heuristic methods. In addition to human-designed heuristic packing policies (e.g. first- or best- fit), there has been interest over the last decade in the automatic generation of policies. One of the main limitations of some previously-used policy representations is the trade-off between locality and granularity in the associated search space. In this article, we adopt an interpolation-based representation which has the jointly-desirable properties of being sparse and continuous (i.e. exhibits good genotype-to-phenotype locality). In contrast to previous approaches, the policy space is searchable via real-valued optimization methods. Packing policies using five different interpolation methods are comprehensively compared against a range of existing methods from the literature, and it is determined that the proposed method scales to larger instances than those in the literature
Bounded space on-line variable-sized bin packing
In this paper we consider the fc-bounded space on-line bin packing problem. Some efficient approximation algorithms are described and analyzed. Selecting either the smallest or the largest available bin size to start a new bin as items arrive turns out to yield a worst-case performance bound of 2. By packing large items into appropriate bins, an efficient approximation algorithm is derived from fc-bounded space on-line bin packing algorithms and its worst-case performance bounds is 1.7 for k > 3
Improving the 1-Bounded Space Algorithms for 2-Dimensional Online Bin Packing
In this paper we study the 1-bounded space of 2-dimensional bin pack-
ing. A sequence of rectangular items arrive one at a time, and the follow-
ing item arrives only after the packing of the previous one, which after
being packed cannot be moved. The bin size is 1 1 and the width and
height of the items are 1. The objective is to minimize the number of
bins used to pack all the items. At any time there is only 1 active bin,
and the previously closed bins cannot be used for any subsequent items.
The new algorithm o ers an improvement of the previous best known
8:84-competitive algorithm to a 6:53-competitive, it also raises the lower
bound from 2:5 to 2:^6
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