Geometric reasoning via internet crowdsourcing

The ability to interpret and reason about shapes is a peculiarly human capability that has proven difficult to reproduce algorithmically. So despite the fact that geometric modeling technology has made significant advances in the representation, display and modification of shapes, there have only been incremental advances in geometric reasoning. For example, although today's CAD systems can confidently identify isolated cylindrical holes, they struggle with more ambiguous tasks such as the identification of partial symmetries or similarities in arbitrary geometries. Even well defined problems such as 2D shape nesting or 3D packing generally resist elegant solution and rely instead on brute force explorations of a subset of the many possible solutions. Identifying economic ways to solving such problems would result in significant productivity gains across a wide range of industrial applications. The authors hypothesize that Internet Crowdsourcing might provide a pragmatic way of removing many geometric reasoning bottlenecks.This paper reports the results of experiments conducted with Amazon's mTurk site and designed to determine the feasibility of using Internet Crowdsourcing to carry out geometric reasoning tasks as well as establish some benchmark data for the quality, speed and costs of using this approach.After describing the general architecture and terminology of the mTurk Crowdsourcing system, the paper details the implementation and results of the following three investigations; 1) the identification of "Canonical" viewpoints for individual shapes, 2) the quantification of "similarity" relationships with-in collections of 3D models and 3) the efficient packing of 2D Strips into rectangular areas. The paper concludes with a discussion of the possibilities and limitations of the approach

TS2PACK: A Two-Level Tabu Search for the Three-dimensional Bin Packing Problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing one

Solving Irregular Strip Packing Problems With Free Rotations Using Separation Lines

Solving nesting problems or irregular strip packing problems is to position polygons in a fixed width and unlimited length strip, obeying polygon integrity containment constraints and non-overlapping constraints, in order to minimize the used length of the strip. To ensure non-overlapping, we used separation lines. A straight line is a separation line if given two polygons, all vertices of one of the polygons are on one side of the line or on the line, and all vertices of the other polygon are on the other side of the line or on the line. Since we are considering free rotations of the polygons and separation lines, the mathematical model of the studied problem is nonlinear. Therefore, we use the nonlinear programming solver IPOPT (an algorithm of interior points type), which is part of COIN-OR. Computational tests were run using established benchmark instances and the results were compared with the ones obtained with other methodologies in the literature that use free rotation

Defragmenting the Module Layout of a Partially Reconfigurable Device

Modern generations of field-programmable gate arrays (FPGAs) allow for partial reconfiguration. In an online context, where the sequence of modules to be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of modules leads to progressive fragmentation of the available space, making defragmentation an important issue. We address this problem by propose an online and an offline component for the defragmentation of the available space. We consider defragmenting the module layout on a reconfigurable device. This corresponds to solving a two-dimensional strip packing problem. Problems of this type are NP-hard in the strong sense, and previous algorithmic results are rather limited. Based on a graph-theoretic characterization of feasible packings, we develop a method that can solve two-dimensional defragmentation instances of practical size to optimality. Our approach is validated for a set of benchmark instances.Comment: 10 pages, 11 figures, 1 table, Latex, to appear in "Engineering of Reconfigurable Systems and Algorithms" as a "Distinguished Paper

Towards Zero-Waste Furniture Design

In traditional design, shapes are first conceived, and then fabricated. While this decoupling simplifies the design process, it can result in inefficient material usage, especially where off-cut pieces are hard to reuse. The designer, in absence of explicit feedback on material usage remains helpless to effectively adapt the design -- even though design variabilities exist. In this paper, we investigate {\em waste minimizing furniture design} wherein based on the current design, the user is presented with design variations that result in more effective usage of materials. Technically, we dynamically analyze material space layout to determine {\em which} parts to change and {\em how}, while maintaining original design intent specified in the form of design constraints. We evaluate the approach on simple and complex furniture design scenarios, and demonstrate effective material usage that is difficult, if not impossible, to achieve without computational support

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions

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