Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

by   Arindam Khan, et al.

We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i^th item has width w(i), height h(i), and d nonnegative weights v_1(i), v_2(i), …, v_d(i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j^th 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 6(d+1) and 3(1 + ln(d+1) + ε). 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 2(1+ln((d+4)/2))+ε for GVBP, which improves to 2.919+ε for the special case of d=1. 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.


page 1

page 2

page 3

page 4


Approximation Algorithms for Generalized Multidimensional Knapsack

We study a generalization of the knapsack problem with geometric and vec...

Harmonic Algorithms for Packing d-dimensional Cuboids Into Bins

We explore approximation algorithms for the d-dimensional geometric bin ...

Approximation Algorithms for Rectangle Packing Problems (PhD Thesis)

In rectangle packing problems we are given the task of placing axis-alig...

Improved Analysis of two Algorithms for Min-Weighted Sum Bin Packing

We study the Min-Weighted Sum Bin Packing problem, a variant of the clas...

Approximation algorithms for the square min-sum bin packing problem

In this work, we study the square min-sum bin packing problem (SMSBPP), ...

Approximation Algorithms for ROUND-UFP and ROUND-SAP

We study ROUND-UFP and ROUND-SAP, two generalizations of the classical B...

Maximum Coverage with Cluster Constraints: An LP-Based Approximation Technique

Packing problems constitute an important class of optimization problems,...

Please sign up or login with your details

Forgot password? Click here to reset