1,602 research outputs found
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 matrix into a product of a nonnegative matrix and a nonnegative matrix . Restricted NMF
requires in addition that the column spaces of and coincide. Finding
the minimal inner dimension 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 always has a restricted NMF of minimal inner
dimension whose factors and are also rational. We show that this holds
for matrices of rank at most and we exhibit a rank- matrix for which
and 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
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