15 research outputs found
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
Lower bound for 3-batched bin packing
Abstract In this paper we will consider a special relaxation of the well-known online bin packing problem. In a batched bin packing problem (BBPP)–defined by Gutin et al. (2005)–the elements come in batches and one batch is available for packing in a given time. If we have K ≥ 2 batches then we denote the problem by K -BBPP. In Gutin et al. (2005) the authors gave a 1.3871 … lower bound for the asymptotic competitive ratio (ACR) of any on-line 2 -BBBP algorithm. In this paper we investigate the 3-BBPP, and we give 1.51211 … lower bound for its ACR
Tight bounds for NF-based bounded-space online bin packing algorithms
In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1/2. Using an NF-based online algorithm the authors proved an ACR of 13/9 = 1.44 … for any given buffer size not less than 1. They also gave a lower bound of 4/3 = 1.33 … for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243, and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of h∞ (r) for the r-parametric problem when the buffer capacity is 3. Since h∞ (2) = 1.42312 …, our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r. © Springer Science+Business Media, LLC 2017
Online Two-Dimensional Vector Packing with Advice
We consider the online two-dimensional vector packing problem, showing a
lower bound of on the competitive ratio of any {\sc AnyFit} strategy for
the problem. We provide strategies with competitive ratio
and
logarithmic advice, for any instance where all the input vectors are restricted
to have angles in the range , for
and
and
logarithmic advice, for any instance where all the input vectors are restricted
to have angles in the range , for
. In addition, we give a -competitive strategy also
using logarithmic advice for the unrestricted vectors case. These results
should be contrasted to the currently best competitive strategy, FirstFit,
having competitive ratio~.Comment: 15 pages, 4 figures. This an extended version of an article published
in "Algorithms and Complexity. CIAC 2021." Lecture Notes in Computer Science,
vol 12701. Springer, https://doi.org/10.100