303 research outputs found
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
Primitive Zonotopes
We introduce and study a family of polytopes which can be seen as a
generalization of the permutahedron of type . We highlight connections
with the largest possible diameter of the convex hull of a set of points in
dimension whose coordinates are integers between and , and with the
computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the
computational complexity of multicriteria matroid optimization was adde
Sweeping an oval to a vanishing point
Given a convex region in the plane, and a sweep-line as a tool, what is best
way to reduce the region to a single point by a sequence of sweeps? The problem
of sweeping points by orthogonal sweeps was first studied in [2]. Here we
consider the following \emph{slanted} variant of sweeping recently introduced
in [1]: In a single sweep, the sweep-line is placed at a start position
somewhere in the plane, then moved continuously according to a sweep vector
(not necessarily orthogonal to the sweep-line) to another parallel end
position, and then lifted from the plane. The cost of a sequence of sweeps is
the sum of the lengths of the sweep vectors. The (optimal) sweeping cost of a
region is the infimum of the costs over all finite sweeping sequences for that
region. An optimal sweeping sequence for a region is one with a minimum total
cost, if it exists. Another parameter of interest is the number of sweeps.
We show that there exist convex regions for which the optimal sweeping cost
cannot be attained by two sweeps. This disproves a conjecture of Bousany,
Karker, O'Rourke, and Sparaco stating that two sweeps (with vectors along the
two adjacent sides of a minimum-perimeter enclosing parallelogram) always
suffice [1]. Moreover, we conjecture that for some convex regions, no finite
sweeping sequence is optimal. On the other hand, we show that both the 2-sweep
algorithm based on minimum-perimeter enclosing rectangle and the 2-sweep
algorithm based on minimum-perimeter enclosing parallelogram achieve a approximation in this sweeping model.Comment: 9 pages, 4 figure
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
On the geometric dilation of closed curves, graphs, and point sets
The detour between two points u and v (on edges or vertices) of an embedded
planar graph whose edges are curves is the ratio between the shortest path in
in the graph between u and v and their Euclidean distance. The maximum detour
over all pairs of points is called the geometric dilation. Ebbers-Baumann,
Gruene and Klein have shown that every finite point set is contained in a
planar graph whose geometric dilation is at most 1.678, and some point sets
require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower
bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem
of packing and covering the plane by circular disks.
The proof relies on halving pairs, pairs of points dividing a given closed
curve C in two parts of equal length, and their minimum and maximum distances h
and H. Additionally, we analyze curves of constant halving distance (h=H),
examine the relation of h to other geometric quantities and prove some new
dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal
submission; it includes additional material from a conference submission
(ref. [6] in the paper
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization
Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem
IST Austria Thesis
This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars
Partitioning a Polygon Into Small Pieces
We study the problem of partitioning a given simple polygon into a
minimum number of polygonal pieces, each of which has bounded size. We give
algorithms for seven notions of `bounded size,' namely that each piece has
bounded area, perimeter, straight-line diameter, geodesic diameter, or that
each piece must be contained in a unit disk, an axis-aligned unit square or an
arbitrarily rotated unit square.
A more general version of the area problem has already been studied. Here we
are, in addition to , given positive real values such that
the sum equals the area of . The goal is to partition
into exactly pieces such that the area of is .
Such a partition always exists, and an algorithm with running time has
previously been described, where is the number of corners of . We give
an algorithm with optimal running time . For polygons with holes, we
get running time .
For the other problems, it seems out of reach to compute optimal partitions
for simple polygons; for most of them, even in extremely restricted cases such
as when is a square. We therefore develop -approximation algorithms
for these problems, which means that the number of pieces in the produced
partition is at most a constant factor larger than the cardinality of a minimum
partition. Existing algorithms do not allow Steiner points, which means that
all corners of the produced pieces must also be corners of . This has the
disappointing consequence that a partition does often not exist, whereas our
algorithms always produce useful partitions. Furthermore, an optimal partition
without Steiner points may require pieces for polygons where a
partition consisting of just pieces exists when Steiner points are allowed.Comment: 32 pages, 24 figure
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