Recent progress in the field of robotic manipulation has generated interest in fully automatic object packing in warehouses. This paper proposes a formulation of the packing problem that is tailored to the automated warehousing domain. Besides minimizing waste space inside a container, the problem requires stability of the object pile during packing and the feasibility of the robot motion executing the placement plans. To address this problem, a set of constraints are formulated, and a constructive packing pipeline is proposed to solve for these constraints. The pipeline is able to pack geometrically complex, non-convex objects with stability while satisfying robot constraints. In particular, a new 3D positioning heuristic called Heightmap-Minimization heuristic is proposed, and heightmaps are used to speed up the search. Experimental evaluation of the method is conducted with a realistic physical simulator on a dataset of scanned real-world items, demonstrating stable and high-quality packing plans compared with other 3D packing methods.
Popular variations of the cutting and packing problem include the bin and strip packing problem, the knapsack problem, the container loading problem, and others. Most existing research on cutting and packing handles floating 2D and 3D rectilinear objects under the non-overlapping constraints. Under some settings, such problems can be formulated and solved to optimally using the exact algorithms. One example of these state-of-the-art exact algorithms is the solution to the 3D bin packing problem using branch and bound, proposed by @cite_9 @cite_10 , whose work is further extended by many including @cite_20 and @cite_6 . The exact algorithms, although capable of finding the optimal solution if infinite time is spent, are strongly NP-hard @cite_26 and do not guarantee optimal results within a reasonable amount of time, especially when a large number of instances are involved @cite_10 . Therefore, heuristic methods and metaheuristic approaches have been developed over the years, such as the popular “Bottom-Left (BL)” heuristic @cite_5 and the Best-Fit-Decreasing heuristic @cite_23 .
@cite_26 @cite_9 @cite_6 @cite_23 @cite_5 @cite_10 @cite_20
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We consider the two-dimensional geometric knapsack problem defined as follows. Given a collection of rectangular axis-parallel items with weights, we want to find a maximum weight subset of the items that can be packed into a rectangular knapsack, i.e., which can be assigned positions within the knapsack such that the items are pairwise non-overlapping. The goal is to compute the optimal collection of items together with a feasible packing. The problem arises naturally in several applications, and various special cases of the problem have been studied. For the general case the best known result is a (2 + e)-approximation algorithm, while the only hardness result is NP-hardness. Our main result is a (1 + e)-approximation algorithm that runs in quasi-polynomial time, provided that the input data consists of (quasi-)polynomially bounded integers. We achieve this result in the setting with and without allowing rotation of the items. Our key technical contribution is to show the existence of a partition for the knapsack into a small number of rectangular boxes. Intuitively, this partition describes the segmentation of the knapsack in some near-optimal solution into large items, areas containing items that are high and thin, and areas containing items that are small and wide. Handling the interaction between these three types of items is a core bottleneck in the design of approximation algorithms for the problem and our partition allows to control this at only marginal cost. In particular, it is so powerful that we do not even need to round the sizes of the items, which is a canonical step in algorithms for geometric knapsack, geometric bin-packing, etc. Finally, we present new algorithms for two-dimensional knapsack and two-dimensional bin-packing under (1 + e)-resource augmentation with a substantially better running time dependence on e (although at the expense of requiring quasi-polynomial time). The objective value of our computed solutions is as good as the optimum without additional resources. Here we exploit a recent result showing the existence of certain low-complexity low weight separators for sets of non-overlapping rectangles. We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + ) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17 9 + ) < 1.89 approximation, which improves to (558 325 + ) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + ) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3 2 + )-approximation for 2DKR, which improves to (4 3 + ) in the cardinality case. Abstract Cutting and packing problems appear under various names in literature, e.g. cutting stock or trim loss problem, bin or strip packing problem, vehicle, pallet or container loading problem, nesting problem, knapsack problem etc. The paper develops a consistent and systematic approach for a comprehensive typology integrating the various kinds of problems. The typology is founded on the basic logical structure of cutting and packing problems. The purpose is to unify the different use of notions in the literature and to concentrate further research on special types of problems. The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit.