New Approximability Results for 2-Dimensional Packing Problems
نویسندگان
چکیده
The strip packing problem is to pack a set of rectangles into a strip of fixed width and minimum length. We present asymptotic polynomial time approximation schemes for this problem without and with 90o rotations. The additive constant in the approximation ratios of both algorithms is 1, improving on the additive term in the approximation ratios of the algorithm by Kenyon and Rémila (for the problem without rotations) and Jansen and van Stee (for the problem with rotations), both of which have a larger additive constant O(1/ε), ε > 0. The algorithms were derived from the study of the rectangle packing problem: Given a set R of rectangles with positive profits, the goal is to find and pack a maximum profit subset of R into a unit size square bin [0, 1] × [0, 1]. We present algorithms that for any value ǫ > 0 find a subset R′ ⊆ R of rectangles of total profit at least (1 − ǫ)OPT , where OPT is the profit of an optimum solution, and pack them (either without rotations or with 90o rotations) into the augmented bin [0, 1]× [0, 1+ ǫ].
منابع مشابه
A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability
We present a new lemma stating that, given an arbitrary packing of a set of rectangles into a larger rectangle, a “structured” packing of nearly the same set of rectangles exists. This lemma has several implications on the approximability of 2-dimensional packing problems. In this paper, we use it to show the existence of a polynomial-time approximation scheme for 2-dimensional geometric knapsa...
متن کاملOn Approximating Four Covering and Packing Problems
In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from fullsibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their appr...
متن کاملApproximation Hardness for Small Occurrence Instances of NP-Hard Problems
The paper contributes to the systematic study (started by Berman and Karpinski) of explicit approximability lower bounds for small occurrence optimization problems. We present parametrized reductions for some packing and covering problems, including 3-Dimensional Matching, and prove the best known inapproximability results even for highly restricted versions of them. For example, we show that i...
متن کاملAbstract: Packing rectangular shapes into a rectangular space is one of the most important discussions on Cutting & Packing problems (C;P) such as: cutting problem, bin-packing problem and distributor's pallet loading problem, etc. Assume a set of rectangular pieces with specific lengths, widths and utility values. Also assume a rectangular packing space with specific width and length. The obj...
متن کاملIterative Packing for Demand and Hypergraph Matching
Iterative rounding has enjoyed tremendous success in elegantly resolving open questions regarding the approximability of problems dominated by covering constraints. Although iterative rounding methods have been applied to packing problems, no single method has emerged that matches the effectiveness and simplicity afforded by the covering case. We offer a simple iterative packing technique that ...
متن کامل