Irrational guards are sometimes needed

In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon such that the guards together can see the whole polygon. We say that a guard at position $x$ sees a point $y$ if the line segment $xy$ is fully contained in the polygon. Despite an extensive study of the art gallery problem, it remained an open question whether there are polygons given by integer coordinates that require guard positions with irrational coordinates in any optimal solution. We give a positive answer to this question by constructing a monotone polygon with integer coordinates that can be guarded by three guards only when we allow to place the guards at points with irrational coordinates. Otherwise, four guards are needed. By extending this example, we show that for every $n$, there is polygon which can be guarded by $3n$ guards with irrational coordinates but need $4n$ guards if the coordinates have to be rational. Subsequently, we show that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution.Comment: 18 pages 10 Figure

Heuristic algorithms for the min-max edge 2-coloring problem

In multi-channel Wireless Mesh Networks (WMN), each node is able to use multiple non-overlapping frequency channels. Raniwala et al. (MC2R 2004, INFOCOM 2005) propose and study several such architectures in which a computer can have multiple network interface cards. These architectures are modeled as a graph problem named \emph{maximum edge $q$-coloring} and studied in several papers by Feng et. al (TAMC 2007), Adamaszek and Popa (ISAAC 2010, JDA 2016). Later on Larjomaa and Popa (IWOCA 2014, JGAA 2015) define and study an alternative variant, named the \emph{min-max edge $q$-coloring}. The above mentioned graph problems, namely the maximum edge $q$-coloring and the min-max edge $q$-coloring are studied mainly from the theoretical perspective. In this paper, we study the min-max edge 2-coloring problem from a practical perspective. More precisely, we introduce, implement and test four heuristic approximation algorithms for the min-max edge $2$-coloring problem. These algorithms are based on a \emph{Breadth First Search} (BFS)-based heuristic and on \emph{local search} methods like basic \emph{hill climbing}, \emph{simulated annealing} and \emph{tabu search} techniques, respectively. Although several algorithms for particular graph classes were proposed by Larjomaa and Popa (e.g., trees, planar graphs, cliques, bi-cliques, hypergraphs), we design the first algorithms for general graphs. We study and compare the running data for all algorithms on Unit Disk Graphs, as well as some graphs from the DIMACS vertex coloring benchmark dataset.Comment: This is a post-peer-review, pre-copyedit version of an article published in International Computing and Combinatorics Conference (COCOON'18). The final authenticated version is available online at: http://www.doi.org/10.1007/978-3-319-94776-1_5

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition

Consensus Paper: Cerebellum and Emotion

Over the past three decades, insights into the role of the cerebellum in emotional processing have substantially increased. Indeed, methodological refinements in cerebellar lesion studies and major technological advancements in the field of neuroscience are in particular responsible to an exponential growth of knowledge on the topic. It is timely to review the available data and to critically evaluate the current status of the role of the cerebellum in emotion and related domains. The main aim of this article is to present an overview of current facts and ongoing debates relating to clinical, neuroimaging, and neurophysiological findings on the role of the cerebellum in key aspects of emotion. Experts in the field of cerebellar research discuss the range of cerebellar contributions to emotion in nine topics. Topics include the role of the cerebellum in perception and recognition, forwarding and encoding of emotional information, and the experience and regulation of emotional states in relation to motor, cognitive, and social behaviors. In addition, perspectives including cerebellar involvement in emotional learning, pain, emotional aspects of speech, and neuropsychiatric aspects of the cerebellum in mood disorders are briefly discussed. Results of this consensus paper illustrate how theory and empirical research have converged to produce a composite picture of brain topography, physiology, and function that establishes the role of the cerebellum in many aspects of emotional processing

Competitive Tendering In The Netherlands: Central Planning Or Functional Specifications?

Institute of Transport and Logistics Studies. Faculty of Economics and Business. The University of Sydne

The mystery of the cerebellum: clues from experimental and clinical observations

Abstract The cerebellum has a striking homogeneous cytoarchitecture and participates in both motor and non-motor domains. Indeed, a wealth of evidence from neuroanatomical, electrophysiological, neuroimaging and clinical studies has substantially modified our traditional view on the cerebellum as a sole calibrator of sensorimotor functions. Despite the major advances of the last four decades of cerebellar research, outstanding questions remain regarding the mechanisms and functions of the cerebellar circuitry. We discuss major clues from both experimental and clinical studies, with a focus on rodent models in fear behaviour, on the role of the cerebellum in motor control, on cerebellar contributions to timing and our appraisal of the pathogenesis of cerebellar tremor. The cerebellum occupies a central position to optimize behaviour, motor control, timing procedures and to prevent body oscillations. More than ever, the cerebellum is now considered as a major actor on the scene of disorders affecting the CNS, extending from motor disorders to cognitive and affective disorders. However, the respective roles of the mossy fibres, the climbing fibres, cerebellar cortex and cerebellar nuclei remains unknown or partially known at best in most cases. Research is now moving towards a better definition of the roles of cerebellar modules and microzones. This will impact on the management of cerebellar disorders