Optimal packing and covering in the plane are NP-complete . (1981)

by R J Fowler, M S Paterson, S L Tanimoto
Venue:Information Processing Letters,
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Approximation schemes for covering and packing problems in image processing and VLSI

by Dorit S. Hochbaum, Wolfgang Maass - J. ACM , 1985
"... A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NP-complete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error c> 0, with running time that is polynomial whe ..."
Abstract - Cited by 249 (0 self) - Add to MetaCart
A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NP-complete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error c> 0, with running time that is polynomial when c is fixed. Thougb the polynomiality of these algorithms depends on the degree of approximation e being fixed, they cannot be improved, owing to a negative result stating that there are no fully polynomial approximation schemes for strongly NP-complete problems unless NP = P. The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems. The method of using the technique and how it varies with problem parameters are illustrated. A similar technique, independently devised by B. S. Baker, was shown to be applicable for covering and packing problems on planar graphs.
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The NP-completeness column: an ongoing guide

by David S. Johnson - JOURNAL OF ALGORITHMS , 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freem ..."
Abstract - Cited by 239 (0 self) - Add to MetaCart
This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should
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Incremental Clustering and Dynamic Information Retrieval

by Moses Charikar, Chandra Chekuri, Tomás Feder, Rajeev Motwani , 1997
"... Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retri ..."
Abstract - Cited by 191 (4 self) - Add to MetaCart
Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application, and which should also be useful in other applications. The goal is to efficiently maintain clusters of small diameter as new points are inserted. We analyze several natural greedy algorithms and demonstrate that they perform poorly. We propose new deterministic and randomized incremental clustering algorithms which have a provably good performance. We complement our positive results with lower bounds on the performance of incremental algorithms. Finally, we consider the dual clustering problem where the clusters are of fixed diameter, and the goal is to minimize the number of clusters.
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NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

by Harry B. Hunt, III, Madhav V. Marathe, VENKATESH RADHAKRISHNAN, S. S. Ravi, Daniel J. Rosenkrantz, Richard E. Stearns , 1997
"... We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar gr ..."
Abstract - Cited by 116 (1 self) - Add to MetaCart
We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of -precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for -precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...

Efficient algorithms for geometric optimization

by Pankaj K. Agarwal, Micha Sharir - ACM Comput. Surv , 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear progra ..."
Abstract - Cited by 114 (10 self) - Add to MetaCart
We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

Surface Approximation and Geometric Partitions

by Pankaj K. Agarwal, Subhash Suri - IN PROC. 5TH ACM-SIAM SYMPOS. DISCRETE ALGORITHMS , 1994
"... Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise linear function \Si ..."
Abstract - Cited by 100 (14 self) - Add to MetaCart
Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise linear function \Sigma(x; y) of minimum complexity (that is, a xy-monotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that j\Sigma(x p ; y p ) \Gamma z p j "; for all (x p ; y p ; z p ) 2 S: We prove that the decision version of this problem is NP-Hard . The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise linear surface of size O(K o log K o ), where K o is the complexity of an optimal surface satisfying the constraints of the problem. The technique
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Realistic Input Models for Geometric Algorithms

by Mark de Berg , Matthew J. Katz, A. Frank van der Stappen , Jules Vleugels - IN PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM , 1997
"... Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed t ..."
Abstract - Cited by 93 (21 self) - Add to MetaCart
Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed that take advantage of these properties. This can lead to algorithms that are provably efficient in realistic situations. We obtain some fundamental results in this research direction. In particular, we have the following results. ffl We show the relations between various models that have been proposed in the literature. ffl For several of these models, we give algorithms to compute the model parameter(s) for a given scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. ffl As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scenes often encountered in ...

Discrete Mobile Centers

by Jie Gao, Leonidas J. Guibas, John Hershberger, Li Zhang, An Zhu - Discrete and Computational Geometry , 2001
"... We propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered a ..."
Abstract - Cited by 92 (13 self) - Add to MetaCart
We propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. The algorithm can be implemented without exact knowledge of the node positions, if each node is able to sense its distance to other nodes up to the cluster radius. Such a kinetic clustering can be used in numerous applications where mobile devices must be interconnected into an ad-hoc network to collaboratively perform some tasks. 1
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Label Placement by Maximum Independent Set in Rectangles

by Pankaj K. Agarwal , Marc Van Kreveld , Subhash Suri - COMPUTATIONAL GEOMETRY , 1998
"... Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)-factor approximation of the maximum subset in a set of n arbitrary axis-pa ..."
Abstract - Cited by 92 (6 self) - Add to MetaCart
Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)-factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can find a 2-approximation in O(n logn) time. Extending this result, we obtain a (1 + l/k)-approximation in time O(n logn + n 2k-1) time, for any integer k/> 1. © 1998 Elsevier Science B.V. All rights reserved.

Exact and Approximation Algorithms for Clustering

by Pankaj K. Agarwal , Cecilia M. Procopiuc , 1997
"... In this paper we present a n O(k1�1=d) time algorithm for solving the k-center problem in R d, under L1 and L2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete k-center problem, as well. We also describe a simple (1 +)-approximation algorithm for the k-center pr ..."
Abstract - Cited by 79 (6 self) - Add to MetaCart
In this paper we present a n O(k1�1=d) time algorithm for solving the k-center problem in R d, under L1 and L2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete k-center problem, as well. We also describe a simple (1 +)-approximation algorithm for the k-center problem, with running time O(n log k) + (k = ) O(k1�1=d). Finally, we present a n O(k1�1=d) time algorithm for solving the L-capacitated k-center problem, provided that L = (n=k 1�1=d) or L = O(1). We conclude with a simple approximation algorithm for the L-capacitated k-center problem.