Orthogonal Packings in Two Dimensions

Abstract

Abstract. We consider problems of packing an arbitrary collection of rectangular pieces into an open-ended, rectangular bin so as to minimize the height achieved by any piece. This problem has numerous applications in operations research and studies of computer operation. We devise efficient approximation algorithms, study their limitations, and derive worst-case bounds on the performance of the packings they produce.Key words, two-dimensional packing, bin packing, resource constrained scheduling 1. Introduction. Efficiently packing sets of rectangular figures into a given rectangular area is a problem with widespread application in operations research. Thus, one is inclined to attribute the scarcity of results on this problem, and others of its type, to inherent difficulty rather than to lack of importance. Motivated by the intractability of these problems, we define and analyze certain approximation algorithms. These algorithms are natural in the sense that they would probably be among the first to occur to anyone wishing to design simple, fast procedures for determining easily computed packings. The analysis of these algorithms leads to bounds on the performance of approximate packings relative to the best achievable.

Two Level Job-Scheduling Strategies for a Computational Grid

“…The LSF has been shown to be a 3-approximation [2]. Some results about asymptotic performance ratio of different strategies for this problem and improvements are presented in [1,4,5].…”

“…Before the time t c the kth node is filled at least half (the property of the BLD algorithm) [2], hence …”

Scheduling Moldable BSP Tasks

“…The basic idea is to select, in a first step, an allocation (the number of processors allocated to each task) and then solve the resulting non-moldable scheduling problem, which is a classical multiprocessor scheduling problem. As far as the makespan criterion is concerned, this problem is identical to a 2-dimensional strip-packing problem [1,4]. It is clear that applying an approximation of guarantee λ for the non-moldable problem on the allocation of an optimal solution provides the same guarantee λ for the moldable problem.…”

A Tabu Search Algorithm with Direct Representation for Strip Packing

“…Based on this permutation representation, several greedy placement heuristics have been investigated for the 2D-SPP. BLF (Bottom Left Fill) is such a heuristic [27]. Basically, BLF places each object at the left-most and lowest possible free area.…”

“…In local search algorithms, the initial solution s 0 specifies where the search begins in S. TSD uses the BLF procedure [27] to construct s 0 , where the π permutation orders the r i ∈ R first by decreasing width, and, second, by decreasing height if necessary (randomly last). We employed this decoder / order since previous experiments suggested that the BLF placement algorithm usually outperforms other φ decoders, see [28,31] for instance.…”