Approximating Smallest Containers for Packing Three-dimensional Convex Objects

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard so that we cannot expect to find exact polynomial time algorithms. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimal volume containers for the objects described

Closing the Gap for Pseudo-Polynomial Strip Packing

Two-dimensional packing problems are a fundamental class of optimization problems and Strip Packing is one of the most natural and famous among them. Indeed it can be defined in just one sentence: Given a set of rectangular axis parallel items and a strip with bounded width and infinite height, the objective is to find a packing of the items into the strip minimizing the packing height. We speak of pseudo-polynomial Strip Packing if we consider algorithms with pseudo-polynomial running time with respect to the width of the strip. It is known that there is no pseudo-polynomial time algorithm for Strip Packing with a ratio better than 5/4 unless P = NP. The best algorithm so far has a ratio of 4/3 + epsilon. In this paper, we close the gap between inapproximability result and currently known algorithms by presenting an algorithm with approximation ratio 5/4 + epsilon. The algorithm relies on a new structural result which is the main accomplishment of this paper. It states that each optimal solution can be transformed with bounded loss in the objective such that it has one of a polynomial number of different forms thus making the problem tractable by standard techniques, i.e., dynamic programming. To show the conceptual strength of the approach, we extend our result to other problems as well, e.g., Strip Packing with 90 degree rotations and Contiguous Moldable Task Scheduling, and present algorithms with approximation ratio 5/4 + epsilon for these problems as well

Improved Pseudo-Polynomial-Time Approximation for Strip Packing

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P!=NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a (7/5+epsilon)-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm (though a PPT approximation scheme might exist). In this paper we make further progress on the PPT approximability of strip packing, by presenting a (4/3+epsilon)-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the "empty space" left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by 90 degrees, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well

Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins

We explore approximation algorithms for the d-dimensional geometric bin packing problem (dBP). Caprara [Caprara, 2008] gave a harmonic-based algorithm for dBP having an asymptotic approximation ratio (AAR) of (T_?)^{d-1} (where T_? ? 1.691). However, their algorithm doesn\u27t allow items to be rotated. This is in contrast to some common applications of dBP, like packing boxes into shipping containers. We give approximation algorithms for dBP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm having AAR T_?^d. We next give a more sophisticated harmonic-based algorithm, which we call HGaP_k, having AAR (T_?)^{d-1}(1+?). This gives an AAR of roughly 2.860 + ? for 3BP with rotations, which improves upon the best-known AAR of 4.5. In addition, we study the multiple-choice bin packing problem that generalizes the rotational case. Here we are given n sets of d-dimensional cuboidal items and we have to choose exactly one item from each set and then pack the chosen items. Our algorithms also work for the multiple-choice bin packing problem. We also give fast and simple approximation algorithms for the multiple-choice versions of dD strip packing and dD geometric knapsack

Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given $n$ rectangular items where the $i^{\textrm{th}}$ item has width $w(i)$, height $h(i)$, and $d$ nonnegative weights $v_1(i), v_2(i), \ldots, v_{d}(i)$. Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all $j \in [d]$, the sum of the $j^{\textrm{th}}$ weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well studied in practice, surprisingly, approximation algorithms for this problem have rarely been explored. We first obtain two simple algorithms for GVBP having asymptotic approximation ratios $6(d+1)$ and $3(1 + \ln(d+1) + \varepsilon)$. We then extend the Round-and-Approx (R&A) framework [Bansal-Khan, SODA'14] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of $2(1+\ln((d+4)/2))+\varepsilon$ for GVBP, which improves to $2.919+\varepsilon$ for the special case of $d=1$. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids