60,101 research outputs found
New results on stabbing segments with a polygon
We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loftier and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard. (C) 2014 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author's final draft
On -Guarding Thin Orthogonal Polygons
Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point
to guard a point if and only if the minimum axis-aligned rectangle spanned
by and is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of -guards is polynomial for tree polygons, but the
run-time was . We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with holes or thickness , becoming fixed-parameter tractable in
.Comment: 18 page
Towards a theory of automated elliptic mesh generation
The theory of elliptic mesh generation is reviewed and the fundamental problem of constructing computational space is discussed. It is argued that the construction of computational space is an NP-Complete problem and therefore requires a nonstandard approach for its solution. This leads to the development of graph-theoretic, combinatorial optimization and integer programming algorithms. Methods for the construction of two dimensional computational space are presented
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
Isoperimetric and stable sets for log-concave perturbations of Gaussian measures
Let be an open half-space or slab in endowed with
a perturbation of the Gaussian measure of the form
, where and is a smooth concave
function depending only on the signed distance from the linear hyperplane
parallel to . In this work we follow a variational approach to
show that half-spaces perpendicular to uniquely minimize the
weighted perimeter in among sets enclosing the same weighted volume.
The main ingredient of the proof is the characterization of half-spaces
parallel or perpendicular to as the unique stable sets with
small singular set and null weighted capacity. Our methods also apply for
, which produces in particular the classification of
stable sets in Gauss space and a new proof of the Gaussian isoperimetric
inequality. Finally, we use optimal transport to study the weighted minimizers
when the perturbation term is concave and possibly non-smooth.Comment: final version, to appear in Analysis and Geometry in Metric Space
- …