Motivated by applications for robust guarding, we consider a variant of the multiple-watchmen problem that ensures that every point within a polygon P is seen from more than one direction: we search for two routes W₁,W₂, such that every point p ∈ P is contained in a segment w₁w₂ ⊆ P such that w₁ ∈ W₁ and w₂ ∈ W₂. We call such routes segment watchman routes. We show that finding the two routes that are optimal with respect to the min-max criterion is weakly NP-hard even in simple polygons, and that finding the routes that are optimal with respect to the min-sum criterion is NP-hard in polygons with holes. Moreover, we present sufficient conditions for routes to be segment watchman routes, and provide a polynomial-time 2-approximation under both the min-max criterion and the min-sum criterion, both in simple polygons. Finally, we show how to generalize our results for k watchmen.

Abstract We study the Art Gallery Problem under k-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the polyomino has length at most k. In this paper, we show that the VC dimension of this problem is 3 in simple polyominoes, and 4 in polyominoes with holes. Furthermore, we provide a reduction from Planar Monotone 3Sat, thereby showing that the problem is -complete even in thin polyominoes (i.e., polyominoes that do not a contain a $$2\times 2$$ 2 × 2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a $$3\times 3$$ 3 × 3 block of cells) for all $$k\in {\mathbb {N}}$$ k ∈ N .

Spiral Galaxies is a pencil-and-paper puzzle played on a grid of unit squares: given a set of points called centers , the goal is to partition the grid into polyominoes such that each polyomino contains exactly one center and is 180 ∘ rotationally symmetric about its center. We show that this puzzle is NP-complete, ASP-complete, and #P-complete even if (a) all solutions to the puzzle have rectangles for polyominoes; or (b) the polyominoes are required to be rectangles and all solutions to the puzzle have just 1 × 1 , 1 × 3 , and 3 × 1 rectangles. The proof for the latter variant also implies NP/ASP/#P-completeness of finding a noncrossing perfect matching in distance-2 grid graphs where edges connect vertices of Euclidean distance 2. Moreover, we prove NP-completeness of the design problem of minimizing the number of centers such that there exists a set of galaxies that exactly cover a given shape.

The art gallery problem (AGP) is one of the classical problems in computational geometry. It asks for the minimum number of guards required to achieve visibility coverage of a given polygon. The AGP is well-known to be NP-hard even in restricted cases. In this paper, we consider the AGP with fading (AGPF): A polygonal region is to be illuminated with light sources such that every point is illuminated with at least a global threshold, light intensity decreases over distance, and we seek to minimize the total energy consumption. Choosing fading exponents of zero, one, and two are equivalent to the AGP, laser scanner applications, and natural light, respectively. We present complexity results as well as a negative solvability result. Still, we propose two practical algorithms for AGPF with fixed light positions (e.g. vertex guards) independent of the fading exponent, which we demonstrate to work well in practice. One is based on a discrete approximation, the other on non-linear programming by means of simplex-partitioning strategies. The former approach yields a fully polynomial-time approximation scheme for the AGPF with fixed light positions. The latter approach obtains better results in our experimental evaluation.

We present an application of Integer Programming to the design of arrival routes for aircraft in a Terminal Maneuvering Area (TMA). We generate operationally feasible merge trees of curvature-constrained routes, using two optimization criteria: (1) total length of the tree, and (2) distance flown along the tree paths. The output routes guarantee that the overall traffic pattern in the TMA can be monitored by air traffic controllers; in particular, we keep merge points for arriving aircraft well separated, and we exclude conflicts between arriving and departing aircraft. We demonstrate the feasibility of our method by experimenting with arrival routes for a runway at Arlanda airport in the Stockholm TMA. Our approach can easily be extended in several ways, e.g., to ensure that the routes avoid no-fly zones.

No abstract available.

In this video, we illustrate how one of the classical areas of computational geometry has gained in practical relevance, which in turn gives rise to new, fascinating geometric problems. In particular, we demonstrate how the robot platform IRMA3D can produce high-resolution, virtual 3D environments, based on a limited number of laser scans. Computing an optimal set of scans amounts to solving an instance of the Art Gallery Problem (AGP): Place a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded.

The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NP-hard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constant-factor approximation algorithms nor exact solution methods are known. We present a primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and—in case of convergence and integrality—ends with an optimal solution. We describe our implementation and give results for an assortment of polygons, including non-orthogonal polygons with holes.

In recent years, tremendous progress has been made in understanding the dynamics of vehicle traffic flow and traffic congestion by interpreting traffic as a multiparticle system. This helps to explain the onset and persistence of many undesired phenomena, for example, traffic jams. It also reflects the apparent helplessness of drivers in traffic, who feel like passive particles that are pushed around by exterior forces; one of the crucial aspects is the inability to communicate and coordinate with other traffic participants. We present distributed methods for solving these fundamental problems, employing modern wireless, ad-hoc, multi-hop networks. The underlying idea is to use these capabilities as the basis for self-organizing methods for coordinating data collection and processing, recognizing traffic phenomena, and changing their structure by coordinated behavior. The overall objective is a multi-level approach that reaches from protocols for local wireless communication, data dissemination, pattern recognition, over hierarchical structuring and coordinated behavior, all the way to large-scale traffic regulation. In this article, we describe three types of results: (i) self-organizing and distributed methods for maintaining and collecting data (using our concept of Hovering Data Clouds ); (ii) adaptive data dissemination for traffic information systems; (iii) methods for self-recognition of traffic jams. We conclude by describing higher-level aspects of our work.

We study a problem related to computer vision: How can a field of sensors compute higher-level properties of observed objects deterministically in sublinear time, without accessing a central authority? This issue is not only important for real-time processing of images, but lies at the very heart of understanding how a brain may be able to function. In particular, we consider a quadratic field of n "smart pixels" on a video chip that observe a B/W image. Each pixel can exchange low-level information with its immediate neighbors. We show that it is possible to compute the centers of gravity along with a principal component analysis of all connected components of the black grid graph in time O(sqrt(n)), by developing appropriate distributed protocols that are modeled after sweepline methods. Our method is not only interesting from a philosophical and theoretical point of view, it is also useful for actual applications for controling a robot arm that has to seize objects on a moving belt. We describe details of an implementation on an FPGA; the code has also been turned into a hardware design for an application-specific integrated circuit (ASIC).

Protocols and applications that rely on unicast and multicast communication are well accepted and still gain more and more popularity. However, these communication paradigms are not optimal for a class of wireless applications where communication partners neither establish specific relationships nor need roles like client and server between each other before data exchange. Applications we have in mind deal with up to several thousands of peers as autonomous wireless network nodes. Nodes communicate events like traffic accidents in a local region or information of common interest to a larger group of network nodes. Intermediate nodes forward or rather "gossip" information like in a social communication model, comparable to the news of the big fire of Rome in neronian times travelling through Europe and finally reaching villages in rural areas. The challenge of such a concept is to find efficient local rules, which balance communication with respect to bandwidth usage, latency of data, and data delivery ratio. We introduce the promising application AutoNomos - a decentralized traffic information system - which is well suited for the evaluation of such a data dissemination protocol. Next, we present our new approach calledAutoCastthat is well optimized and self-adaptable towards various dynamic topologies. We compareAutoCastagainst the theoretical optimum and existing data dissemination protocols. Finally, simulations will demonstrate the efficiency of the approach.

Many complex structures in our modern world exist independent of the individual entities they are composed of, giving them an "organic" quality. Important examples include traffic phenomena, e.g., traffic jams; despite of strong efforts over many years, centralized computing has been unable to deal with the resulting problems in a satisfactory manner. With the growing power of sensing devices and wireless communication, participants in traffic are no longer restricted to display passive, particle-like behavior; instead, local data exchange makes it technically feasible to aim for decentralized coordination between cars. One fundamental concept for making use of these possibilities comprises Hovering Data Clouds, which consist of relevant information that is kept by ever-changing carriers; a prototypical scenario arises in a traffic jam, where data is maintained by passing it on to newly arriving cars. In this study, we present algorithmic methods for this concept. This paper is part of project AutoNomos1(www.auto- nomos.de), which aims at traffic control in a decentralized manner.