Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

On Restricted Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. Restricted NMF requires in addition that the column spaces of $M$ and $W$ coincide. Finding the minimal inner dimension $d$ is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closely related to a question about the nature of minimal probabilistic automata, posed by Paz in his seminal 1971 textbook. We use this connection to answer Paz's question negatively, thus falsifying a positive answer claimed in 1974. Furthermore, we investigate whether a rational matrix $M$ always has a restricted NMF of minimal inner dimension whose factors $W$ and $H$ are also rational. We show that this holds for matrices $M$ of rank at most $3$ and we exhibit a rank-$4$ matrix for which $W$ and $H$ require irrational entries.Comment: Full version of an ICALP'16 pape

On the Geometric Interpretation of the Nonnegative Rank

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square

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

Identifying Alternate Optimal Solutions to the Design Approximation Problem in Stock Cutting

The design approximation problem is a well known problem in stock cutting, where, in order to facilitate the optimization techniques used in the cutting process, it is required to approximate complex designs by simpler ones. Although there are algorithms available to solve this problem, they all suffer from an undesirable feature that they only produce one optimal solution to the problem, and do not identify the complete set of all optimal solutions. The focus of this paper is to study this hitherto unexplored aspect of the problem: specifically, the case is considered in which both the design and the parent material are convex shapes, and some essential properties of all optimal solutions to the design approximation problem are ascertained. These properties are then used to devise two efficient schemes to identify the set of all optimal solutions to the problem. Finally, the recovery of a desired optimal approximation from the identified sets of optimal solutions, is discussed

On the geometric interpretation of the nonnegative rank

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.nonnegative rank, restricted nonnegative rank, nested polytopes, computational complexity, computational geometry, extended formulations, linear Euclidean distance matrices.

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Manhattan orbifolds

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L_1 (or equivalently L_infinity) metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised since the previous version, and a new example has been adde