Improving the 1-Bounded Space Algorithms for 2-Dimensional Online Bin Packing

In this paper we study the 1-bounded space of 2-dimensional bin pack- ing. A sequence of rectangular items arrive one at a time, and the follow- ing item arrives only after the packing of the previous one, which after being packed cannot be moved. The bin size is 1 1 and the width and height of the items are 1. The objective is to minimize the number of bins used to pack all the items. At any time there is only 1 active bin, and the previously closed bins cannot be used for any subsequent items. The new algorithm o ers an improvement of the previous best known 8:84-competitive algorithm to a 6:53-competitive, it also raises the lower bound from 2:5 to 2:^6

Improving the 1-Bounded Space Algorithms for 2-Dimensional Online Bin Packing

In this paper we study the 1-bounded space of 2-dimensional bin pack- ing. A sequence of rectangular items arrive one at a time, and the follow- ing item arrives only after the packing of the previous one, which after being packed cannot be moved. The bin size is 1 1 and the width and height of the items are 1. The objective is to minimize the number of bins used to pack all the items. At any time there is only 1 active bin, and the previously closed bins cannot be used for any subsequent items. The new algorithm o ers an improvement of the previous best known 8:84-competitive algorithm to a 6:53-competitive, it also raises the lower bound from 2:5 to 2:^6

Online algorithms for 1-space bounded multi dimensional bin packing and hypercube packing

In this paper, we study 1-space bounded multi-dimensional bin packing and hypercube packing. A sequence of items arrive over time, each item is a d-dimensional hyperbox (in bin packing) or hypercube (in hypercube packing), and the length of each side is no more than 1. These items must be packed without overlapping into d-dimensional hypercubes with unit length on each side. In d-dimensional space, any two dimensions i and j define a space P ij. When an item arrives, we must pack it into an active bin immediately without any knowledge of the future items, and 90 {ring operator}-rotation on any plane P ij is allowed. The objective is to minimize the total number of bins used for packing all these items in the sequence. In the 1-space bounded variant, there is only one active bin for packing the current item. If the active bin does not have enough space to pack the item, it must be closed and a new active bin is opened. For d-dimensional bin packing, an online algorithm with competitive ratio 4 d is given. Moreover, we consider d-dimensional hypercube packing, and give a 2 d+1-competitive algorithm. These two results are the first study on 1-space bounded multi dimensional bin packing and hypercube packing. © 2012 The Author(s).published_or_final_versionSpringer Open Choice, 28 May 201

A Survey of Classical and Recent Results in Bin Packing Problem

In the classical bin packing problem one receives a sequence of n items 1, 2,..., n with sizes s1, s2, . . . ,sn where each item has a fixed size in (0, 1]. One needs to find a partition of the items into sets of size1, called bins, so that the number of sets in the partition is minimized and the sum of the sizes of the pieces assigned to any bin does not exceed its capacity. This combinatorial optimization problem which is NP hard has many variants as well as online and offline versions of the problem. Though the problem is well studied and numerous results are known, there are many open problems. Recently bin packing has gained renewed attention in as a tool in the area of cloud computing. We give a survey of different variants of the problem like 2D bin packing, strip packing, bin packing with rejection and emphasis on recent results. The thesis contains a discussion of a newly claimed tight result for First Fit Decreasing by Dosa et.al. as well as various new versions of the problem by Epstein and others