Almost optimal set covers in finite VC-dimension
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A deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of the authors' cover is at most a factor ofO(d log(dc)) from the optimal size,c.Abstract:
We give a deterministic polynomial-time method for finding a set cover in a set system (X, ?) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous ?(log?X?) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.read more
Citations
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Journal ArticleDOI
The Multiplicative Weights Update Method: A Meta-Algorithm and Applications
TL;DR: A simple meta-algorithm is presented that unifies many of these disparate algorithms and derives them as simple instantiations of the meta-Algorithm.
Survey of Polygonal Surface Simplification Algorithms
Paul S. Heckbert,Michael Garland +1 more
TL;DR: Methods for simplifying and approximating polygonal surfaces from computer graphics, computer vision, cartography, computational geometry, and other fields are classified, summarized, and compared both practically and theoretically.
Proceedings ArticleDOI
Simplification envelopes
Jonathan D. Cohen,Amitabh Varshney,Dinesh Manocha,Greg Turk,Hans Weber,Pankaj K. Agarwal,Frederick P. Brooks,William Wright +7 more
TL;DR: The idea of simplification envelopes for generating a hierarchy of level-of-detail approximations for a given polygonal model is proposed and guarantees that all points of an approximation are within a user-specifiable distance from the original model.
Book ChapterDOI
Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems
TL;DR: The problem of computing a minimum set of solutions that approximates within a specified accuracy the Pareto curve of a multi-objective optimization problem was studied in this article, where it was shown that it is NP-hard to do better than 2.
Journal ArticleDOI
Connected sensor cover: self-organization of sensor networks for efficient query execution
TL;DR: The notion of a connected sensor cover is developed and a centralized approximation algorithm that constructs a topology involving a near-optimal connected sensors cover is designed, which proves that the size of the constructed topology is within an O(log n) factor of the optimal size.
References
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TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
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TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.