2,234 research outputs found
Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing
We study the generalized multidimensional bin packing problem (GVBP) that
generalizes both geometric packing and vector packing. Here, we are given
rectangular items where the item has width , height
, and nonnegative weights . Our
goal is to get an axis-parallel non-overlapping packing of the items into
square bins so that for all , the sum of the
weight of items in each bin is at most 1. This is a natural problem arising in
logistics, resource allocation, and scheduling. Despite being well studied in
practice, surprisingly, approximation algorithms for this problem have rarely
been explored.
We first obtain two simple algorithms for GVBP having asymptotic
approximation ratios and . We then
extend the Round-and-Approx (R&A) framework [Bansal-Khan, SODA'14] to wider
classes of algorithms, and show how it can be adapted to GVBP. Using more
sophisticated techniques, we obtain better approximation algorithms for GVBP,
and we get further improvement by combining them with the R&A framework. This
gives us an asymptotic approximation ratio of
for GVBP, which improves to for the special case of .
We obtain further improvement when the items are allowed to be rotated. We also
present algorithms for a generalization of GVBP where the items are high
dimensional cuboids
Optimal online bounded space multidimensional packing
We solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space. We show that it is optimal among bounded space algorithms for any dimension . Its asymptotic performance ratio is , where is the asymptotic performance ratio of the one-dimensional algorithm harm. A modified version of this algorithm for the case where all items are hypercubes is also shown to be optimal. Its asymptotic performance ratio is sublinear in . Additionally, for the special case of packing squares in two-dimensional bins, we present a new unbounded space online algorithm with asymptotic performance ratio of at most . We also present an approximation algorithm for the offline problem with approximation ratio of . This improves upon all earlier approximation algorithms for this problem, including the algorithm from Caprara, Packing 2-dimensional bins in harmony, Proc. 43rd FOCS, 2002
Vector Bin Packing with Multiple-Choice
We consider a variant of bin packing called multiple-choice vector bin
packing. In this problem we are given a set of items, where each item can be
selected in one of several -dimensional incarnations. We are also given
bin types, each with its own cost and -dimensional size. Our goal is to pack
the items in a set of bins of minimum overall cost. The problem is motivated by
scheduling in networks with guaranteed quality of service (QoS), but due to its
general formulation it has many other applications as well. We present an
approximation algorithm that is guaranteed to produce a solution whose cost is
about times the optimum. For the running time to be polynomial we
require and . This extends previous results for vector
bin packing, in which each item has a single incarnation and there is only one
bin type. To obtain our result we also present a PTAS for the multiple-choice
version of multidimensional knapsack, where we are given only one bin and the
goal is to pack a maximum weight set of (incarnations of) items in that bin
Improved approximation bounds for Vector Bin Packing
In this paper we propose an improved approximation scheme for the Vector Bin
Packing problem (VBP), based on the combination of (near-)optimal solution of
the Linear Programming (LP) relaxation and a greedy (modified first-fit)
heuristic. The Vector Bin Packing problem of higher dimension (d \geq 2) is not
known to have asymptotic polynomial-time approximation schemes (unless P = NP).
Our algorithm improves over the previously-known guarantee of (ln d + 1 +
epsilon) by Bansal et al. [1] for higher dimensions (d > 2). We provide a
{\theta}(1) approximation scheme for certain set of inputs for any dimension d.
More precisely, we provide a 2-OPT algorithm, a result which is irrespective of
the number of dimensions d.Comment: 15 pages, 3 algorithm
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