8,272 research outputs found
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
A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game
We prove a tight lower bound on the asymptotic performance ratio of
the bounded space online -hypercube bin packing problem, solving an open
question raised in 2005. In the classic -hypercube bin packing problem, we
are given a sequence of -dimensional hypercubes and we have an unlimited
number of bins, each of which is a -dimensional unit hypercube. The goal is
to pack (orthogonally) the given hypercubes into the minimum possible number of
bins, in such a way that no two hypercubes in the same bin overlap. The bounded
space online -hypercube bin packing problem is a variant of the
-hypercube bin packing problem, in which the hypercubes arrive online and
each one must be packed in an open bin without the knowledge of the next
hypercubes. Moreover, at each moment, only a constant number of open bins are
allowed (whenever a new bin is used, it is considered open, and it remains so
until it is considered closed, in which case, it is not allowed to accept new
hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448]
showed that is and , and conjectured that
it is . We show that is in fact . To
obtain this result, we elaborate on some ideas presented by those authors, and
go one step further showing how to obtain better (offline) packings of certain
special instances for which one knows how many bins any bounded space algorithm
has to use. Our main contribution establishes the existence of such packings,
for large enough , using probabilistic arguments. Such packings also lead to
lower bounds for the prices of anarchy of the selfish -hypercube bin packing
game. We present a lower bound of for the pure price of
anarchy of this game, and we also give a lower bound of for
its strong price of anarchy
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
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