ONLINE BIN STRETCHING is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for ONLINE BIN STRETCHING with a stretching factor of 1.5 for any number of bins. We build o...

The vector bin packing problem (VBP) is a generalization of bin packing with multiple constraints. In this problem we are required to pack items, represented by p-dimensional vectors, into as few bins as possible. The multiple-choice vector bin packing (MVBP) is a variant of the VBP in which bins have several types and items have several incarnations. We present an exact method, based on an arc...

The main capability of the dual-frequency profiling radar proposed for deployment on the Aerosol/Cloud/Ecosystems (ACE) mission [1] is its ability to allow retrievals of vertical profiles of water content and mean hydrometeor size. At the center of this capability is the notion that measuring the radar reflectivity at two sufficiently different frequencies should allow the retrieval of two wate...

This article explores the central antagonisms at work within Alī nāma of Rabīʿ, a fifth/eleventh-century Persian epic by Twelver Shīʿī poet narrating deeds ʿAlī b. Abī Ṭālib, in particular battles Camel and Ṣiffīn. It gives an analysis this poem its twin contexts imamology poetry, illustrating how these fields are variously drawn upon, combined challenged poem’s unique narrative imam, arguing t...

The following online bin packing problem is considered: Items with integer sizes are given and variable sized bins arrive online. A bin must be used if there is still an item remaining which fits in it when the bin arrives. The goal is to minimize the total size of all the bins used. Previously, a lower bound of 5 4 on the competitive ratio of this problem was achieved using jobs of size S and ...

In this paper, we study the following knapsack problem: Given a list of squares with profits, we are requested to pack a sublist of them into a rectangular bin (not a unit square bin) to make profits in the bin as large as possible. We first observe there is a Polynomial Time Approximation Scheme (PTAS) for the problem of packing weighted squares into rectangular bins with large resources, then...

The bin packing problem (BP) consists in finding the minimum number of bins necessary to pack a set of items so that the total size of the items in each bin does not exceed the bin capacity C. The bin capacity is common for all the bins. This problem can be solved in Constraint Programming (CP) by introducing one placement variable xi for each item and one load variable lj for each bin. The Pac...

Suppose we sequentially put n balls into n bins. If we put each ball into a random bin then the heaviest bin will contain ∼ logn/ log logn balls with high probability. However, Azar, Broder, Karlin and Upfal [SIAM J. Comput. 29 (1999) 180–200] showed that if each time we choose two bins at random and put the ball in the least loaded bin among the two, then the heaviest bin will contain only ∼ l...

Bin packing is a well-known NP-Hard optimization problem which has several real applications. Classical bin packing (BPP) is a simple model where all bins are identical. However, the variable sized bin packing problem (VSBPP) is a generalization of the bin packing problem where bins of different capacities are available for packing a set of items having different weights. The objective is to pa...