Facets for Art Gallery Problems

The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The problem is known to be NP-hard, and its inherent continuous structure (with both the set of points that need to be guarded and the set of points that can be used for guarding being uncountably infinite) makes it difficult to apply a straightforward formulation as an Integer Linear Program. We use an iterative primal-dual relaxation approach for solving AGP instances to optimality. At each stage, a pair of LP relaxations for a finite candidate subset of primal covering and dual packing constraints and variables is considered; these correspond to possible guard positions and points that are to be guarded. Particularly useful are cutting planes for eliminating fractional solutions. We identify two classes of facets, based on Edge Cover and Set Cover (SC) inequalities. Solving the separation problem for the latter is NP-complete, but exploiting the underlying geometric structure, we show that large subclasses of fractional SC solutions cannot occur for the AGP. This allows us to separate the relevant subset of facets in polynomial time. We also characterize all facets for finite AGP relaxations with coefficients in {0, 1, 2}. Finally, we demonstrate the practical usefulness of our approach. Our cutting plane technique yields a significant improvement in terms of speed and solution quality due to considerably reduced integrality gaps as compared to the approach by Kr\"oller et al.Comment: 29 pages, 18 figures, 1 tabl

Alzheimer’s Disease under the Purview of Graph Theory Centric Genetic Networks

Notice that the synapsis of brain is a form of communication. As communication demands connectivity, it is not a surprise that "graph theory" is a fastest growing area of research in the life sciences. It attempts to explain the connections and communication between networks of neurons. Alzheimer’s disease (AD) progression in brain is due to a deposition and development of amyloid plaque and the loss of communication between nerve cells. Graph/network theory can provide incredible insights into the incorrect wiring leading to memory loss in a progressive manner. Network in AD is slanted towards investigating the intricate patterns of interconnections found in the pathogenesis of brain. Here, we see how the notions of graph/network theory can be prudently exploited to comprehend the Alzheimer’s disease. We begin with introducing concepts of graph/network theory as a model for specific genetic hubs of the brain regions and cellular signalling. We begin with a brief introduction of prevalence and causes of AD followed by outlining its genetic and signalling pathogenesis. We then present some of the network-applied outcome in assessing the disease-signalling interactions, signal transduction of protein-protein interaction, disturbed genetics and signalling pathways as compelling targets of pathogenesis of the disease.</em

Crew Scheduling for Netherlands Railways: "destination: customer"

: In this paper we describe the use of a set covering model with additional constraints for scheduling train drivers and conductors for the Dutch railway operator NS Reizigers. The schedules were generated according to new rules originating from the project "Destination: Customer" ("Bestemming: Klant" in Dutch). This project is carried out by NS Reizigers in order to increase the quality and the punctuality of its train services. With respect to the scheduling of drivers and conductors, this project involves the generation of efficient and acceptable duties with a high robustness against the transfer of delays of trains. A key issue for the acceptability of the duties is the included amount of variation per duty. The applied set covering model is solved by dynamic column generation techniques, Lagrangean relaxation and powerful heuristics. The model and the solution techniques are part of the TURNI system, which is currently used by NS Reizigers for carrying out several analyses concerning the required capacities of the depots. The latter are strongly influenced by the new rules

Graph Theory Applications to Comprehend Epidemics Spread of a Disease

Theory of Graphs could offer a plenty to enrich the analysis and modelling to generate datasets out of the systems and processes regarding the spread of a disease that affects humans, animals, plants, crops etc., In this paper first we show graphs can serve as a model for cattle movements from one farm to another. Second, we give a crisp explanation regarding disease transmission models on contact graphs/networks. It is possible to indicate how a regular tree exhibits relations among graph structure and the infectious disease spread and how certain properties of it akin to diameter and density of graph, affect the duration of an outbreak. Third, we elaborate on the presence of a suitable environment for exploiting several streams of data such as genetic temporal and spatial to locate case clusters one dependent on the other of a disease that is infectious. Here a graph for each stream of data joining all cases that are created with pairwise distance among them as edge weights and altered by omitting exceeding distances of a cut-off assigned that relies on already existing assumptions and rate of spread of a disease information. Fourth we provide an overview of epidemiology, disease transmission, fatality rate and clinical features of zoonotic viral infections of epidemic and pandemic magnitude since 2000. Fifth we indicate how the clinical data and virus spread data can be exploited for the creation of health knowledge graph. Graph Theory is an ideal tool to model, predict, form an opinion to devise strategies to quickly arrest the outbreak and minimize the devastating effect of zoonotic viral infections.</p

Restoring Vision through “Project Prakash”: The Opportunities for Merging Science and Service

“So how does this help society?” is a question we are often asked as scientists. The lack of immediate and tangible results cannot be held against a scientific project but statements of future promise in broad and inchoate terms can sometimes pass the benefit-buck indefinitely. There is no incentive against over-stating the benefits, especially when they are hypothetical and lie in the distant future. Few scientists will say their science is not designed to serve society. Yet the proliferation of “potential benefits” in grant proposals and the Discussion sections of research papers, in the absence of tangible translations, can make the service element of science seem like a cliched ritual. Its repetition hollows out its meaning, breeding cynicism about the idea that basic science can be of service

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes

Scheduling divisible loads with time and cost constraints

In distributed computing, divisible load theory provides an important system model for allocation of data-intensive computations to processing units working in parallel. The main task is to define how a computation job should be split into parts, to which processors those parts should be allocated and in which sequence. The model is characterized by multiple parameters describing processor availability in time, transfer times of job parts to processors, their computation times and processor usage costs. The main criteria are usually the schedule length and cost minimization. In this paper, we provide the generalized formulation of the problem, combining key features of divisible load models studied in the literature, and prove its NP-hardness even for unrestricted processor availability windows. We formulate a linear program for the version of the problem with a fixed number of processors. For the case with an arbitrary number of processors, we close the gaps in the study of special cases, developing efficient algorithms for single criterion and bicriteria versions of the problem, when transfer times are negligible

Statistical mechanics of the vertex-cover problem

We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits an coverable-uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy-hard transitions in the typical running time of the algorithms occur. We explain a statistical mechanics approach, which works by mapping VC to a hard-core lattice gas, and then applying techniques like the replica trick or the cavity approach. Using these methods, the phase diagram of VC could be obtained exactly for connectivities $c<e$, where VC is replica symmetric. Recently, this result could be confirmed using traditional mathematical techniques. For $c>e$, the solution of VC exhibits full replica symmetry breaking. The statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for VC. Finally, we describe recent results for VC when studied on other ensembles of finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math. Ge