29,321 research outputs found
Computational Approaches to Lattice Packing and Covering Problems
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice packing-covering
problem. Theoretically our algorithms solve the two problems in any fixed
dimension d in the sense that they approximate optimal covering lattices and
optimal packing-covering lattices within any desired accuracy. Both algorithms
involve semidefinite programming and are based on Voronoi's reduction theory
for positive definite quadratic forms, which describes all possible Delone
triangulations of Z^d.
In practice, our implementations reproduce known results in dimensions d <= 5
and in particular solve the two problems in these dimensions. For d = 6 our
computations produce new best known covering as well as packing-covering
lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our
approach leads to new best known covering lattices. Although we use numerical
methods, we made some effort to transform numerical evidences into rigorous
proofs. We provide rigorous error bounds and prove that some of the new
lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in
Discrete and Computational Geometry, see also
http://fma2.math.uni-magdeburg.de/~latgeo
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
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On a linear program for minimum-weight triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial-time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: For an LP formulation due to Dantzig, Hoffman, and Hu [Math. Programming, 31 (1985), pp. 1-14], (i) the integrality gap is constant, and (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP. © 2014 Society for Industrial and Applied Mathematics
On a Linear Program for Minimum-Weight Triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time
constant-factor approximation algorithm, and a variety of effective polynomial-
time heuristics that, for many instances, can find the exact MWT. Linear
programs (LPs) for MWT are well-studied, but previously no connection was known
between any LP and any approximation algorithm or heuristic for MWT. Here we
show the first such connections: for an LP formulation due to Dantzig et al.
(1985): (i) the integrality gap is bounded by a constant; (ii) given any
instance, if the aforementioned heuristics find the MWT, then so does the LP.Comment: To appear in SICOMP. Extended abstract appeared in SODA 201
Minimum-weight triangulation is NP-hard
A triangulation of a planar point set S is a maximal plane straight-line
graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we
are looking for a triangulation of a given point set that minimizes the sum of
the edge lengths. We prove that the decision version of this problem is
NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the
gadgets is established with computer assistance, using dynamic programming on
polygonal faces, as well as the beta-skeleton heuristic to certify that certain
edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures.
This revision contains a few improvements in the expositio
An ETH-Tight Exact Algorithm for Euclidean TSP
We study exact algorithms for {\sc Euclidean TSP} in . In the
early 1990s algorithms with running time were presented for
the planar case, and some years later an algorithm with
running time was presented for any . Despite significant interest in
subexponential exact algorithms over the past decade, there has been no
progress on {\sc Euclidean TSP}, except for a lower bound stating that the
problem admits no algorithm unless ETH fails. Up to
constant factors in the exponent, we settle the complexity of {\sc Euclidean
TSP} by giving a algorithm and by showing that a
algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201
Higher signature Delaunay decompositions
A Delaunay decomposition is a cell decomposition in R^d for which each cell
is inscribed in a Euclidean ball which is empty of all other vertices. This
article introduces a generalization of the Delaunay decomposition in which the
Euclidean balls in the empty ball condition are replaced by other families of
regions bounded by certain quadratic hypersurfaces. This generalized notion is
adaptable to geometric contexts in which the natural space from which the point
set is sampled is not Euclidean, but rather some other flat semi-Riemannian
geometry, possibly with degenerate directions. We prove the existence and
uniqueness of the decomposition and discuss some of its basic properties. In
the case of dimension d = 2, we study the extent to which some of the
well-known optimality properties of the Euclidean Delaunay triangulation
generalize to the higher signature setting. In particular, we describe a higher
signature generalization of a well-known description of Delaunay decompositions
in terms of the intersection angles between the circumscribed circles.Comment: 25 pages, 6 figure
Geodesic-Preserving Polygon Simplification
Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon by a polygon such that (1)
contains , (2) has its reflex
vertices at the same positions as , and (3) the number of vertices
of is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
and , our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of
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