8 research outputs found
Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins
We explore approximation algorithms for the d-dimensional geometric bin packing problem (dBP). Caprara [Caprara, 2008] gave a harmonic-based algorithm for dBP having an asymptotic approximation ratio (AAR) of (T_?)^{d-1} (where T_? ? 1.691). However, their algorithm doesn\u27t allow items to be rotated. This is in contrast to some common applications of dBP, like packing boxes into shipping containers. We give approximation algorithms for dBP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm having AAR T_?^d. We next give a more sophisticated harmonic-based algorithm, which we call HGaP_k, having AAR (T_?)^{d-1}(1+?). This gives an AAR of roughly 2.860 + ? for 3BP with rotations, which improves upon the best-known AAR of 4.5. In addition, we study the multiple-choice bin packing problem that generalizes the rotational case. Here we are given n sets of d-dimensional cuboidal items and we have to choose exactly one item from each set and then pack the chosen items. Our algorithms also work for the multiple-choice bin packing problem. We also give fast and simple approximation algorithms for the multiple-choice versions of dD strip packing and dD geometric knapsack
Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items
In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA\u2714]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS\u2705] obtained an APTAS for this problem. Let ? be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by ? opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ? ? ? 1.692. Bansal and Khan [SODA\u2714] conjectured that ? = 4/3. The conjecture, if true, will imply a (4/3+?)-approximation algorithm for 2BP. According to convention, for a given constant ? > 0, a rectangle is large if both its height and width are at least ?, and otherwise it is called skewed. We make progress towards the conjecture by showing ? = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on ? was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS
Simplification and Improvement of MMS Approximation
We consider the problem of fairly allocating a set of indivisible goods among
agents with additive valuations, using the popular fairness notion of
maximin share (MMS). Since MMS allocations do not always exist, a series of
works provided existence and algorithms for approximate MMS allocations. The
current best approximation factor, for which the existence is known, is
[Garg and Taki, 2021]. Most of these results
are based on complicated analyses, especially those providing better than
factor. Moreover, since no tight example is known of the Garg-Taki algorithm,
it is unclear if this is the best factor of this approach. In this paper, we
significantly simplify the analysis of this algorithm and also improve the
existence guarantee to a factor of . For small , this provides a noticeable improvement.
Furthermore, we present a tight example of this algorithm, showing that this
may be the best factor one can hope for with the current techniques
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
Existence and Computation of Epistemic EFX Allocations
We consider the problem of allocating indivisible goods among agents in a
fair manner. For this problem, one of the best notions of fairness is
envy-freeness up to any good (EFX). However, it is not known if EFX allocations
always exist. Hence, several relaxations of EFX allocations have been studied.
We propose another relaxation of EFX, called epistemic EFX (EEFX). An
allocation is EEFX iff for every agent , it is possible to shuffle the goods
of the other agents such that agent does not envy any other agent up to any
good. We show that EEFX allocations always exist for additive valuations, and
give a polynomial-time algorithm for computing them. We also show how EEFX is
related to some previously-known notions of fairness.Comment: Edit: include additional authors and some minor change