Facets for Art Gallery Problems

The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The problem is known to be NP-hard, and its inherent continuous structure (with both the set of points that need to be guarded and the set of points that can be used for guarding being uncountably infinite) makes it difficult to apply a straightforward formulation as an Integer Linear Program. We use an iterative primal-dual relaxation approach for solving AGP instances to optimality. At each stage, a pair of LP relaxations for a finite candidate subset of primal covering and dual packing constraints and variables is considered; these correspond to possible guard positions and points that are to be guarded. Particularly useful are cutting planes for eliminating fractional solutions. We identify two classes of facets, based on Edge Cover and Set Cover (SC) inequalities. Solving the separation problem for the latter is NP-complete, but exploiting the underlying geometric structure, we show that large subclasses of fractional SC solutions cannot occur for the AGP. This allows us to separate the relevant subset of facets in polynomial time. We also characterize all facets for finite AGP relaxations with coefficients in {0, 1, 2}. Finally, we demonstrate the practical usefulness of our approach. Our cutting plane technique yields a significant improvement in terms of speed and solution quality due to considerably reduced integrality gaps as compared to the approach by Kr\"oller et al.Comment: 29 pages, 18 figures, 1 tabl

Stock Cutting Of Complicated Designs by Computing Minimal Nested Polygons

This paper studies the following problem in stock cutting: when it is required to cut out complicated designs from parent material, it is cumbersome to cut out the exact design or shape, especially if the cutting process involves optimization. In such cases, it is desired that, as a first step, the machine cut out a relatively simpler approximation of the original design, in order to facilitate the optimization techniques that are then used to cut out the actua1 design. This paper studies this problem of approximating complicated designs or shapes. The problem is defined formally first and then it is shown that this problem is equivalent to the Minima1 Nested Polygon problem in geometry. Some properties of the problem are then shown and it is demonstrated that the problem is related to the Minimal Turns Path problem in geometry. With these results, an efficient approximate algorithm is obtained for the origina1 stock cutting problem. Numerica1 examples are provided to illustrate the working of the algorithm in different cases

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.Comment: 37 pages, 9 figure

Polynomial Meshes: Computation and Approximation

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs

Optimal positioning of irregular shapes in stamping die strip

The nesting of two-dimensional shapes is a common problem, where raw material has to be economically cut. As for the single-pass single-row strip layout, several algorithms, based on established methods, have been proposed. Moreover, it should be noticed that the optimum layout should also consider a few constraints, like grain orientation for subsequent forming operation, correct bridge width, and the commercial roll of metal width in order to make solutions applicable in real industrial environments. Most of the procedures until now shown in literature are quite complex and often ignore these real constraints. They usually make use of sliding techniques and are not able to effectively work with relatively multiple-connected figures. In particular, most of the different proposed procedures are based on the No Fit Polygon (NFP) computation of non-convex polygons, which often generates holes. This work is a proposal for a more efficient method, which can be used in heuristic procedures. In order to overcome some faults of most of the former methods presented in literature, in this paper a new geometric entity called \u201cNo Fit Path\u201d (NFPh) of non-convex polygons is applied. It allows researchers to find solutions of nesting problems even when there are NFP faults due to degenerate solutions. Moreover, the No Fit Path allows researchers to easily read, modify, or share their results, overcoming all those problems arising from the usual large amount of information and from the different origins and formats of the obtained data. Given two non-convex polygons, the algorithm is able to calculate their NFPh very quickly and without any approximation by a polygon clipping method. In this paper a totally automated procedure has been developed. This procedure firstly obtains the \u201cNo Fit Path\u201d (NFPh); secondly, between all the existing positions on the NFPh, the algorithm searches the optimal one, minimizing the global waste. The proposed approach also allows designers to set an optimal orientation of the shapes on the roll of metal, taking account of the grain orientation in order to obtain the best mechanical characteristics for the cut pieces