Complexity of the General Chromatic Art Gallery Problem

In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon $P$. We consider the Chromatic AGP (CAGP), where the guards are colored. As long as $P$ is completely covered, the number of guards does not matter, but guards with overlapping visibility regions must have different colors. This problem has applications in landmark-based mobile robot navigation: Guards are landmarks, which have to be distinguishable (hence the colors), and are used to encode motion primitives, \eg, "move towards the red landmark". Let $\chi_G(P)$, the chromatic number of $P$, denote the minimum number of colors required to color any guard cover of $P$. We show that determining, whether $\chi_G(P) \leq k$ is \NP-hard for all $k \geq 2$. Keeping the number of colors minimal is of great interest for robot navigation, because less types of landmarks lead to cheaper and more reliable recognition

Judd on Phenomena

Donald Judd’s 1964 essay 'Specific Objects' probably remains his most well-known. In it, he described new artworks characterized by, among other features, 'a quality as a whole' instead of conventional 'part-by-part structure,' the 'use of three dimensions' and 'real space' as opposed to depiction, 'new materials [that] aren’t obviously art,' and the unadorned appearance and 'obdurate identity' of materials as they are. Judd held that the 'shape, image, color and surface' of these objects were more 'specific,' that is to say, 'more intense, clear and powerful,' than in previous art. While these positions demonstrate Judd’s subjective preferences as an artist and art critic, they also convey some of the wider debates driving American avant-garde practices in the 1960s, such as the supposed 'insufficiencies of painting and sculpture' as mediums. Art historians tend to find such breadth appealing of course - sweeping statements bring retrospective order to what was actually haphazard and unruly. But Judd knew that you lose much in eliminating complexity for the sake of clarity. He emphasized this point in his earlier essay 'Local History' so as to qualify the more general of his own arguments. 'The history of art and art’s condition at any time are pretty messy,' he declared. 'They should stay that way.

Person re-identification via efficient inference in fully connected CRF

In this paper, we address the problem of person re-identification problem, i.e., retrieving instances from gallery which are generated by the same person as the given probe image. This is very challenging because the person's appearance usually undergoes significant variations due to changes in illumination, camera angle and view, background clutter, and occlusion over the camera network. In this paper, we assume that the matched gallery images should not only be similar to the probe, but also be similar to each other, under suitable metric. We express this assumption with a fully connected CRF model in which each node corresponds to a gallery and every pair of nodes are connected by an edge. A label variable is associated with each node to indicate whether the corresponding image is from target person. We define unary potential for each node using existing feature calculation and matching techniques, which reflect the similarity between probe and gallery image, and define pairwise potential for each edge in terms of a weighed combination of Gaussian kernels, which encode appearance similarity between pair of gallery images. The specific form of pairwise potential allows us to exploit an efficient inference algorithm to calculate the marginal distribution of each label variable for this dense connected CRF. We show the superiority of our method by applying it to public datasets and comparing with the state of the art.Comment: 7 pages, 4 figure

The Dispersive Art Gallery Problem

We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery problem we are given a polygon ? and a real number ?, and want to decide whether ? has a guard set such that every pair of guards in this set is at least a distance of ? apart. In this paper, we study the vertex guard variant of this problem for the class of polyominoes. We consider rectangular visibility and distances as geodesics in the L?-metric. Our results are as follows. We give a (simple) thin polyomino such that every guard set has minimum pairwise distances of at most 3. On the positive side, we describe an algorithm that computes guard sets for simple polyominoes that match this upper bound, i.e., the algorithm constructs worst-case optimal solutions. We also study the computational complexity of computing guard sets that maximize the smallest distance between all pairs of guards within the guard sets. We prove that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP-complete. We were also able to find an optimal dynamic programming approach that computes a guard set that maximizes the minimum pairwise distance between guards in tree-shaped polyominoes, i.e., computes optimal solutions; due to space constraints, details can be found in the full version of our paper [Christian Rieck and Christian Scheffer, 2022]. Because the shapes constructed in the NP-hardness reduction are thin as well (but have holes), this result completes the case for thin polyominoes

Conflict-Free Coloring of Intersection Graphs

A conflict-free k-coloring of a graph G=(V,E) 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\u27s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of n geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in Omega(log n/log log n) and in Omega(sqrt{log n}) for disks or squares of different sizes; it is known for general graphs that the worst case is in Theta(log^2 n). For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two