16,006 research outputs found
On Graph Crossing Number and Edge Planarization
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its
vertices into points of the plane, and its edges into continuous curves,
connecting the images of their endpoints. A crossing in such a drawing is a
point where two such curves intersect. In the Minimum Crossing Number problem,
the goal is to find a drawing of G with minimum number of crossings. The value
of the optimal solution, denoted by OPT, is called the graph's crossing number.
This is a very basic problem in topological graph theory, that has received a
significant amount of attention, but is still poorly understood
algorithmically. The best currently known efficient algorithm produces drawings
with crossings on bounded-degree graphs, while only a
constant factor hardness of approximation is known. A closely related problem
is Minimum Edge Planarization, in which the goal is to remove a
minimum-cardinality subset of edges from G, such that the remaining graph is
planar. Our main technical result establishes the following connection between
the two problems: if we are given a solution of cost k to the Minimum Edge
Planarization problem on graph G, then we can efficiently find a drawing of G
with at most \poly(d)\cdot k\cdot (k+OPT) crossings, where is the maximum
degree in G. This result implies an O(n\cdot \poly(d)\cdot
\log^{3/2}n)-approximation for Minimum Crossing Number, as well as improved
algorithms for special cases of the problem, such as, for example, k-apex and
bounded-genus graphs
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
Coplanar Repeats by Energy Minimization
This paper proposes an automated method to detect, group and rectify
arbitrarily-arranged coplanar repeated elements via energy minimization. The
proposed energy functional combines several features that model how planes with
coplanar repeats are projected into images and captures global interactions
between different coplanar repeat groups and scene planes. An inference
framework based on a recent variant of -expansion is described and fast
convergence is demonstrated. We compare the proposed method to two widely-used
geometric multi-model fitting methods using a new dataset of annotated images
containing multiple scene planes with coplanar repeats in varied arrangements.
The evaluation shows a significant improvement in the accuracy of
rectifications computed from coplanar repeats detected with the proposed method
versus those detected with the baseline methods.Comment: 14 pages with supplemental materials attache
Planar Ultrametric Rounding for Image Segmentation
We study the problem of hierarchical clustering on planar graphs. We
formulate this in terms of an LP relaxation of ultrametric rounding. To solve
this LP efficiently we introduce a dual cutting plane scheme that uses minimum
cost perfect matching as a subroutine in order to efficiently explore the space
of planar partitions. We apply our algorithm to the problem of hierarchical
image segmentation
Labeling Schemes for Bounded Degree Graphs
We investigate adjacency labeling schemes for graphs of bounded degree
. In particular, we present an optimal (up to an additive
constant) adjacency labeling scheme for bounded degree trees.
The latter scheme is derived from a labeling scheme for bounded degree
outerplanar graphs. Our results complement a similar bound recently obtained
for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new
insights for closing the long standing gap for adjacency in trees [Alstrup and
Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree
planar graphs. Finally, we use combinatorial number systems and present an
improved adjacency labeling schemes for graphs of bounded degree with
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Plane-extraction from depth-data using a Gaussian mixture regression model
We propose a novel algorithm for unsupervised extraction of piecewise planar
models from depth-data. Among other applications, such models are a good way of
enabling autonomous agents (robots, cars, drones, etc.) to effectively perceive
their surroundings and to navigate in three dimensions. We propose to do this
by fitting the data with a piecewise-linear Gaussian mixture regression model
whose components are skewed over planes, making them flat in appearance rather
than being ellipsoidal, by embedding an outlier-trimming process that is
formally incorporated into the proposed expectation-maximization algorithm, and
by selectively fusing contiguous, coplanar components. Part of our motivation
is an attempt to estimate more accurate plane-extraction by allowing each model
component to make use of all available data through probabilistic clustering.
The algorithm is thoroughly evaluated against a standard benchmark and is shown
to rank among the best of the existing state-of-the-art methods.Comment: 11 pages, 2 figures, 1 tabl
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