LOCATING GUARDS FOR VISIBILITY COVERAGE OF POLYGONS

We propose heuristics for visibility coverage of a polygon with the fewest point guards. This optimal coverage problem, often called the "art gallery problem", is known to be NP-hard, so most recent research has focused on heuristics and approximation methods. We evaluate our heuristics through experimentation, comparing the upper bounds on the optimal guard number given by our methods with computed lower bounds based on heuristics for placing a large number of visibility-independent "witness points". We give experimental evidence that our heuristics perform well in practice, on a large suite of input data; often the heuristics give a provably optimal result, while in other cases there is only a small gap between the computed upper and lower bounds on the optimal guard number.

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EDGE GUARDS IN STRAIGHT WALKABLE POLYGONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195999000066 ◽

1999 ◽

Vol 09 (01) ◽

pp. 63-79

Author(s):

XUEHOU TAN

Keyword(s):

Lower Bounds ◽

Simple Polygon ◽

Upper Bounds ◽

The Other ◽

Art Gallery ◽

Art Gallery Problem ◽

Monotone Polygons ◽

Polygonal Chains ◽

The Given

We study the art gallery problem restricted to edge guards and straight walkable polygons. An edge guard is the guard that patrols individual edges of the polygon. A simple polygon P is called straight walkable if there are two vertices s and t in P and we can move two points montonically on two polygonal chains of P from s to t, one clockwise and the other counterclockwise, such that two points are always mutually visible. For instance, monotone polygons and spiral polygons are straight walkable. We show that ⌊(n+2)/5⌋ edge guards are always sufficient to watch and n-vertex gallery of this type. Furthermore, we also show that if the given polygon is straight walkable and rectilinear, then ⌊(n+3)/6⌋ edge guards are sufficient. Both of our upper bounds match the known lower bounds.

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Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

Monte Carlo Methods and Applications ◽

10.1515/mcma-2020-2062 ◽

2020 ◽

Vol 26 (2) ◽

pp. 131-161

Author(s):

Florian Bourgey ◽

Stefano De Marco ◽

Emmanuel Gobet ◽

Alexandre Zhou

Keyword(s):

Monte Carlo ◽

Lower Bounds ◽

Asymptotic Optimality ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Independent Random Variables ◽

Outer Function ◽

Control Problems ◽

Multilevel Monte Carlo ◽

Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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ON THE ORDER OF MAGNITUDE OF SUMS OF NEGATIVE POWERS OF INTEGRATED PROCESSES

Econometric Theory ◽

10.1017/s0266466612000503 ◽

2012 ◽

Vol 29 (3) ◽

pp. 642-658 ◽

Cited By ~ 1

Author(s):

Benedikt M. Pötscher

Keyword(s):

Lower Bounds ◽

Random Variables ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Integrated Process ◽

Order Of Magnitude ◽

Image Position ◽

Integrated Processes

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

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Composite Thermoelectrics - Exact Results and Calculational Methods

MRS Proceedings ◽

10.1557/proc-234-39 ◽

1991 ◽

Vol 234 ◽

Cited By ~ 6

Author(s):

David J. Bergman ◽

Ohad Levy

Keyword(s):

Quality Factor ◽

Computer Simulations ◽

Lower Bounds ◽

Theoretical Study ◽

Transport Coefficients ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Exact Results ◽

Thermoelectric Transport

ABSTRACTA theoretical study of composite thermoelectric media has resulted in the development of a number of simple approximations, as well as some exact results. The latter include exact upper and lower bounds on the bulk effective thermoelectric transport coefficients of the composite and upper bounds on the bulk effective thermoelectric quality factor Ze. In particular, as a result of some exact theorems and computer simulations we conclude that Ze can never be greater than the largest value of Z in the different components that make up the composite.

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Computational Approaches for Stochastic Shortest Path on Succinct MDPs

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2018/653 ◽

2018 ◽

Cited By ~ 2

Author(s):

Krishnendu Chatterjee ◽

Hongfei Fu ◽

Amir Goharshady ◽

Nastaran Okati

Keyword(s):

Lower Bounds ◽

Polynomial Time ◽

Shortest Path ◽

Upper Bounds ◽

Decision Processes ◽

Upper And Lower Bounds ◽

Computational Approaches ◽

Stochastic Shortest Path ◽

Markov Decision ◽

Infinite State

We consider the stochastic shortest path (SSP) problem for succinct Markov decision processes (MDPs), where the MDP consists of a set of variables, and a set of nondeterministic rules that update the variables. First, we show that several examples from the AI literature can be modeled as succinct MDPs. Then we present computational approaches for upper and lower bounds for the SSP problem: (a) for computing upper bounds, our method is polynomial-time in the implicit description of the MDP; (b) for lower bounds, we present a polynomial-time (in the size of the implicit description) reduction to quadratic programming. Our approach is applicable even to infinite-state MDPs. Finally, we present experimental results to demonstrate the effectiveness of our approach on several classical examples from the AI literature.

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An efficient algorithm for solving the constellation-to-ground region coverage problem based on longitude strip division

Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University ◽

10.1051/jnwpu/20213940919 ◽

2021 ◽

Vol 39 (4) ◽

pp. 919-929

Author(s):

Zhiming Song ◽

Haidong Liu ◽

Xiaoyu Chen ◽

Maocai Wang

Keyword(s):

Numerical Simulation ◽

Lower Bounds ◽

Efficient Solution ◽

Solution Method ◽

Upper And Lower Bounds ◽

Coverage Problem ◽

Simulation Experiments ◽

Coverage Area ◽

Latitude Range ◽

Continuous Coverage

Aiming at the constellation-to-ground region coverage problem, an efficient solution method called the longitude strip method is proposed. By dividing the ground region into several longitude strips, the latitude range of each strip is computed according to the region. Similarly, concerning the coverage area of the satellite, the latitude range of each strip is also calculated. On this basis, the coverage rate can be obtained by comprehensive statistics. Additionally, the upper and lower bounds for problems of continuous coverage and accumulative coverage are solved. Numerical simulation experiments show the proposed algorithm has higher accuracy and it is also efficient in resolving the ground area with arbitrary shapes.

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Bounds for the Natural Frequencies of a Simply Supported beam Carrying a Uniformly Distributed Axial Load

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1967_009_023_02 ◽

1967 ◽

Vol 9 (2) ◽

pp. 149-156 ◽

Cited By ~ 7

Author(s):

G. Fauconneau ◽

W. M. Laird

Keyword(s):

Lower Bounds ◽

Axial Load ◽

Natural Frequencies ◽

Ritz Method ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Simply Supported Beam ◽

Simply Supported ◽

Distributed Load ◽

Made In

Upper and lower bounds for the eigenvalues of uniform simply supported beams carrying uniformly distributed axial load and constant end load are obtained. The upper bounds were calculated by the Rayleigh-Ritz method, and the lower bounds by a method due to Bazley and Fox. Some results are given in terms of two loading parameters. In most cases the gap between the bounds over their average is less than 1 per cent, except for values of the loading parameters corresponding to the beam near buckling. The results are compared with the eigenvalues of the same beam carrying half of the distributed load lumped at each end. The errors made in the lumping process are very large when the distributed load and the end load are of opposite signs. The results also indicate that the Rayleigh-Ritz upper bounds computed with the eigenfunctions of the unloaded beam as co-ordinate functions are quite accurate.

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Some Refinements and Generalizations of I. Schur Type Inequalities

The Scientific World JOURNAL ◽

10.1155/2014/709358 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Xian-Ming Gu ◽

Ting-Zhu Huang ◽

Wei-Ru Xu ◽

Hou-Biao Li ◽

Liang Li ◽

...

Keyword(s):

Lower Bounds ◽

Recent Literature ◽

Structural Characteristics ◽

Variable Parameter ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Numerical Examples ◽

Schur Inequality

Recently, extensive researches on estimating the value ofehave been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help ofMapleand automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.

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On Proportions of Fit Individuals in Population of Mutation-Based Evolutionary Algorithm with Tournament Selection

Evolutionary Computation ◽

10.1162/evco_a_00210 ◽

2018 ◽

Vol 26 (2) ◽

pp. 269-297 ◽

Cited By ~ 9

Author(s):

Anton V. Eremeev

Keyword(s):

Local Search ◽

Evolutionary Algorithm ◽

Lower Bounds ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Set Cover ◽

Fitness Level ◽

Level Model ◽

Tournament Selection ◽

Tail Bound

In this article, we consider a fitness-level model of a non-elitist mutation-only evolutionary algorithm (EA) with tournament selection. The model provides upper and lower bounds for the expected proportion of the individuals with fitness above given thresholds. In the case of so-called monotone mutation, the obtained bounds imply that increasing the tournament size improves the EA performance. As corollaries, we obtain an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and polynomial upper bounds on the runtime of EAs on the 2-SAT problem and on a family of Set Cover problems proposed by E. Balas.

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PATHWIDTH AND LAYERED DRAWINGS OF TREES

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195904001433 ◽

2004 ◽

Vol 14 (03) ◽

pp. 203-225 ◽

Cited By ~ 23

Author(s):

MATTHEW SUDERMAN

Keyword(s):

Lower Bounds ◽

Linear Time ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Parallel Lines ◽

Number Of Layers ◽

Straight Line ◽

Linear Time Algorithms ◽

Planar Drawing

An h-layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end-vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper h-layer drawing contains only proper edges, a short h-layer drawing contains no long edges, an upright h-layer drawing contains no flat edges, and an unconstrained h-layer drawing contains any type of edge. In this paper, we derive upper and lower bounds on the number of layers required by proper, short, upright, and unconstrained layered drawings of trees. We prove that these bounds are optimal with respect to the pathwidth of the tree being drawn. Finally, we give linear-time algorithms for obtaining layered drawings that match these upper bounds.

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