7,441 research outputs found
Counting Carambolas
We give upper and lower bounds on the maximum and minimum number of geometric
configurations of various kinds present (as subgraphs) in a triangulation of
points in the plane. Configurations of interest include \emph{convex
polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also
consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and
Combinatorics; 18 pages, 13 figure
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
-Stars or On Extending a Drawing of a Connected Subgraph
We consider the problem of extending the drawing of a subgraph of a given
plane graph to a drawing of the entire graph using straight-line and polyline
edges. We define the notion of star complexity of a polygon and show that a
drawing of an induced connected subgraph can be extended with at
most bends per edge, where is the
largest star complexity of a face of and is the size of the
largest face of . This result significantly improves the previously known
upper bound of [5] for the case where is connected. We also show
that our bound is worst case optimal up to a small additive constant.
Additionally, we provide an indication of complexity of the problem of testing
whether a star-shaped inner face can be extended to a straight-line drawing of
the graph; this is in contrast to the fact that the same problem is solvable in
linear time for the case of star-shaped outer face [9] and convex inner face
[13].Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
Range of the first three eigenvalues of the planar Dirichlet Laplacian
We conduct extensive numerical experiments aimed at finding the admissible
range of the ratios of the first three eigenvalues of a planar Dirichlet
Laplacian. The results improve the previously known theoretical estimates of M
Ashbaugh and R Benguria. We also prove some properties of a maximizer of the
ratio .Comment: 21 page; 9 figure
Primitive points in rational polygons
Let be a star-shaped polygon in the plane, with rational
vertices, containing the origin. The number of primitive lattice points in the
dilate is asymptotically Area as
. We show that the error term is both and . Both
bounds extend (to the above class of polygons) known results for the isosceles
right triangle, which appear in the literature as bounds for the error term in
the summatory function for Euler's .Comment: 17 page
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