# Seminars

### Approximation Algorithms for Geometric Packing Problems

Series: M.Tech (Research) Colloquium- ONLINE

Speaker: Mr. Eklavya Sharma M.Tech (Research) Dept.of CSA

Date/Time: Mar 30 16:00:00

Location: Microsoft Teams - ONLINE

Abstract:
We study approximation algorithms for the geometric bin packing problem and its variants. In the two-dimensional geometric bin packing problem (2D GBP), we are given n rectangular items and we have to compute an axis-parallel non-overlapping packing of the items into the minimum number of square bins of side length 1. 2D GBP is an important problem in computer science and operations research arising in logistics, resource allocation, and scheduling.
We first study an extension of 2D GBP called the generalized multidimensional bin packing problem (GVBP). Here each item i additionally has d nonnegative weights v_1(i), v_2(i), â€¦, v_d(i) associated with it. Our goal is to compute an axis-parallel non-overlapping packing of the items into bins so that for all j âˆˆ [d], the sum of the jth weight of items in each bin is at most 1. 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 (AARs) 6(d+1) and 3(1 + ln(d+1) + Îµ). 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 algorithms for GVBP having an AAR of 2(1+ln((d+4)/2))+Îµ, which improves to 2.919+Îµ for the special case of d=1.
Next, we explore approximation algorithms for the d-dimensional geometric bin packing problem (dD GBP). Caprara (MOR 2008) gave a harmonic-based algorithm for dD GBP having an AAR of 1.69104^(d-1). However, their algorithm doesnt 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 dD GBP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm, called fullh_k, having an AAR of 1.69104^d. We next give a more sophisticated harmonic-based algorithm, which we call hgap_k, having an AAR of (1+eps)1.69104^(d-1). This gives an AAR of roughly 2.860 + Îµ 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 fullh_k and hgap_k 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. These algorithms have AARs 1.69104^(d-1) and (1-Îµ)3^(-d), respectively.
A rectangle is said to be Î´-thin if it has width at most Î´ or height at most Î´. When Î´ is a small constant (i.e., close to 0), we give an APTAS for 2D GBP when all rectangles are Î´-thin. On the other hand, general 2D GBP is APX-hard. This shows that hard instances arise due to items that are large in both dimensions.
A packing of rectangles into a bin is said to be guillotine-separable iff we can use a sequence of end-to-end cuts to separate the items from each other. The asymptotic price of guillotinability (APoG) is the maximum value of opt_G(I)/opt(I) for large opt(I), where opt(I) and opt_G(I) are the minimum number of bins and the minimum number of guillotine-separable bins, respectively, needed to pack I. Computing lower and upper bounds on APoG is an important problem, since proving an upper bound smaller than 1.5 would beat the state-of-the-art algorithm for 2D GBP. The best-known upper bound is 1.69104 and the best-known lower bound is 4/3. We analyze this problem for the special case of Î´-thin rectangles, where Î´ is a small constant (i.e., close to 0). We give a roughly 4/3-asymptotic-approximate algorithm for 2D GBP for this case, which proves an upper-bound of roughly 4/3 on APoG for Î´-thin rectangles. We also prove a matching lower-bound of 4/3. This shows that hard examples for upper-bounding APoG include items that are large in both dimensions.

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