We solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space. To achieve this, we introduce a new technique for classifying the items to be packed. We show that our algorithm is optimal among bounded space algorithms for any dimension d > 1. Its asymptotic performance ratio is (Π∞) , where Π∞ ≈ 1.691 is the asymptotic ...

In this paper we study the Multiple Strip Packing (MSP) problem, a generalization of the well-known Strip Packing problem. For a given set of rectangles, r1, . . . , rn, with heights and widths ≤ 1, the goal is to find a non-overlapping orthogonal packing without rotations into k ∈ N strips [0, 1]× [0,∞), minimizing the maximum of the heights. We present an approximation algorithm with absolute...

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min{cx : x ∈ Z+, Ax ≥ a, Bx ≤ b, x ≤ d}. We give a bicriteria-approximation algorithm that, given ε ∈ (0, 1], finds a solution of cost O(ln(m)/ε) times optimal, meeting the covering constraints (Ax ≥ a) and multiplicity constraints (x ≤ d), and satisfying Bx ≤ (1 + ε)b + β, wher...

In the past decade, many parameterized algorithms were developed for packing problems. Our goal is to obtain tradeoffs that improve the running times of these algorithms at the cost of computing approximate solutions. Consider a packing problem for which there is no known algorithm with approximation ratio α, and a parameter k. If the value of an optimal solution is at least k, we seek a soluti...

In the k-dimensional packing problem, we are given a set Z = (b,, b,, . . , b,} of k-dimensional boxes and a k-dimensional box B with unit length in each of the first k 1 dimensions and unbounded length in the kth dimension. Each box bi is represented by a k-tuple b, = (xj’), . . . , xik-l), xjk’> E (0, ilk-’ X (0, a), where x(I) denotes its length in the jth dimension, 1 <j < k. We are asked t...

Simchi-Levi (Naval Res. Logist. 41 (1994) 579–585) proved that the famous bin packing algorithms FF and BF have an absolute worst-case ratio of no more than 4 , and FFD and BFD have an absolute worst-case ratio of 3 2 , respectively. These algorithms run in time O(n log n). In this paper, we provide a linear time constant space (number of bins kept during the execution of the algorithm is const...

The dense packing of particulate assemblies is an enduring theoretical and practical problem. In spite of its apparent simplicity, this problem still remains challenging. Recent progress in the performance of computers boosted the development and realization of a number of effective packing algorithms. Yet, because of the complexity of the problem, many of existing methods have difficulties in ...

The relative worst-order ratio is a measure of the quality of online algorithms. In contrast to the competitive ratio, this measure compares two online algorithms directly instead of using an intermediate comparison with an optimal offline algorithm. In this paper, we apply the relative worst-order ratio to online algorithms for several common variants of the bin packing problem. We mainly cons...