1,582 research outputs found
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
A Practical Algorithm with Performance Guarantees for the Art Gallery Problem
Given a closed simple polygon P, we say two points p,q see each other if the segment seg(p,q) is fully contained in P. The art gallery problem seeks a minimum size set G ? P of guards that sees P completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods. As the art gallery problem is ? ?-complete, it seems unlikely to avoid algebraic methods, for any exact algorithm, without additional assumptions.
In this paper, we introduce the notion of vision-stability. In order to describe vision-stability consider an enhanced guard that can see "around the corner" by an angle of ? or a diminished guard whose vision is by an angle of ? "blocked" by reflex vertices. A polygon P has vision-stability ? if the optimal number of enhanced guards to guard P is the same as the optimal number of diminished guards to guard P. We will argue that most relevant polygons are vision-stable. We describe a one-shot vision-stable algorithm that computes an optimal guard set for vision-stable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision-stable polygon. We implemented an iterative vision-stable algorithm and show its practical performance is slower, but comparable with other state-of-the-art algorithms. The practical implementation can be found at: https://github.com/simonheng/AGPIterative. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord c of a polygon, we denote by n(c) the number of vertices visible from c. The chord-visibility width (cw(P)) of a polygon is the maximum n(c) over all possible chords c. The set of vision-stable polygons admit an FPT algorithm when parameterized by the chord-visibility width. Furthermore, the one-shot algorithm runs in FPT time when parameterized by the number of reflex vertices
Why Chromatic Imaging Matters
During the last two decades, the first generation of beam combiners at the
Very Large Telescope Interferometer has proved the importance of optical
interferometry for high-angular resolution astrophysical studies in the near-
and mid-infrared. With the advent of 4-beam combiners at the VLTI, the u-v
coverage per pointing increases significantly, providing an opportunity to use
reconstructed images as powerful scientific tools. Therefore, interferometric
imaging is already a key feature of the new generation of VLTI instruments, as
well as for other interferometric facilities like CHARA and JWST. It is thus
imperative to account for the current image reconstruction capabilities and
their expected evolutions in the coming years. Here, we present a general
overview of the current situation of optical interferometric image
reconstruction with a focus on new wavelength-dependent information,
highlighting its main advantages and limitations. As an Appendix we include
several cookbooks describing the usage and installation of several state-of-the
art image reconstruction packages. To illustrate the current capabilities of
the software available to the community, we recovered chromatic images, from
simulated MATISSE data, using the MCMC software SQUEEZE. With these images, we
aim at showing the importance of selecting good regularization functions and
their impact on the reconstruction.Comment: Accepted for publication in Experimental Astronomy as part of the
topical collection: Future of Optical-infrared Interferometry in Europ
Weighted and filtered mutual information: A Metric for the automated creation of panoramas from views of complex scenes
To contribute a novel approach in the field of image registration and panorama creation, this algorithm foregoes any scene knowledge, requiring only modest scene overlap and an acceptable amount of entropy within each overlapping view. The weighted and filtered mutual information (WFMI) algorithm has been developed for multiple stationary, color, surveillance video camera views and relies on color gradients for feature correspondence. This is a novel extension of well-established maximization of mutual information (MMI) algorithms. Where MMI algorithms are typically applied to high altitude photography and medical imaging (scenes with relatively simple shapes and affine relationships between views), the WFMI algorithm has been designed for scenes with occluded objects and significant parallax variation between non-affine related views. Despite these typically non-affine surveillance scenarios, searching in the affine space for a homography is a practical assumption that provides computational efficiency and accurate results, even with complex scene views. The WFMI algorithm can perfectly register affine views, performs exceptionally well with near-affine related views, and in complex scene views (well beyond affine constraints) the WFMI algorithm provides an accurate estimate of the overlap regions between the views. The WFMI algorithm uses simple calculations (vector field color gradient, Laplacian filtering, and feature histograms) to generate the WFMI metric and provide the optimal affine relationship. This algorithm is unique when compared to typical MMI algorithms and modern registration algorithms because it avoids almost all a priori knowledge and calculations, while still providing an accurate or useful estimate for realistic scenes. With mutual information weighting and the Laplacian filtering operation, the WFMI algorithm overcomes the failures of typical MMI algorithms in scenes where complex or occluded shapes do not provide sufficiently large peaks in the mutual information maps to determine the overlap region. This work has currently been applied to individual video frames and it will be shown that future work could easily extend the algorithm into utilizing motion information or temporal frame registrations to enhance scenes with smaller overlap regions, lower entropy, or even more significant parallax and occlusion variations between views
Colouring Polygon Visibility Graphs and Their Generalizations
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time
Minimizing Turns in Watchman Robot Navigation: Strategies and Solutions
The Orthogonal Watchman Route Problem (OWRP) entails the search for the
shortest path, known as the watchman route, that a robot must follow within a
polygonal environment. The primary objective is to ensure that every point in
the environment remains visible from at least one point on the route, allowing
the robot to survey the entire area in a single, continuous sweep. This
research places particular emphasis on reducing the number of turns in the
route, as it is crucial for optimizing navigation in watchman routes within the
field of robotics. The cost associated with changing direction is of
significant importance, especially for specific types of robots. This paper
introduces an efficient linear-time algorithm for solving the OWRP under the
assumption that the environment is monotone. The findings of this study
contribute to the progress of robotic systems by enabling the design of more
streamlined patrol robots. These robots are capable of efficiently navigating
complex environments while minimizing the number of turns. This advancement
enhances their coverage and surveillance capabilities, making them highly
effective in various real-world applications.Comment: 6 pages, 3 figure
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