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### An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee [chapter]

Stephan Eidenbenz, Peter Widmayer
2001 Lecture Notes in Computer Science
The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP -hard, even for polygons without holes  .  ...  We propose a polynomial-time approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(log n), where n is the number of vertices in the input  ...  We want to thank anonymous referees for pointing us to [9, 13] and for additional suggestions.  ...

### An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee

Stephan J. Eidenbenz, Peter Widmayer
2003 SIAM journal on computing (Print)
The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP -hard, even for polygons without holes  .  ...  We propose a polynomial-time approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(log n), where n is the number of vertices in the input  ...  We want to thank anonymous referees for pointing us to [9, 13] and for additional suggestions.  ...

### Page 6462 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews
approximation algorithm for minimum convex cover with logarithmic performance guarantee.  ...  Furthermore, we show that MINIMUM CONVEX COVER is APX-hard; i.e., there ex- ists a constant 06 >0 such that no polynomial-time algorithm can achieve an approximation ratio of 1 +06.  ...

### On the Reflexivity of Point Sets [chapter]

Esther M. Arkin, Sándor P. Fekete, Ferran Hurtado, Joseph S. B. Mitchell, Marc Noy, Vera Sacristán, Saurabh Sethia
2001 Lecture Notes in Computer Science
We prove that it is NPcomplete to determine the convex cover or the convex partition number, and we give logarithmic-approximation algorithms for determining each.  ...  We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general).  ...  Algorithmic results that give logarithmic approximations for convex cover number, convex partitioning number, and (Steiner) reflex-ivity.  ...

### Optimal simplification of polygonal chain for rendering

Lilian Buzer
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07
For a given polygonal chain, we study the min-# problem, which consists in finding an approximate and ordered subchain with a minimum number of vertices.  ...  Previous approaches simplify the input chain relative to an approximation criterion which minimizes the gap between the original chain and the simplified subchain.  ...  CONCLUSION Our algorithm computes an approximation with a minimum number of vertices and our hybrid criterion guarantees that the approximation retains the shape of the original chain.  ...

### Maximum Clique and Minimum Clique Partition in Visibility Graphs [chapter]

Stephan Eidenbenz, Christoph Stamm
2000 Lecture Notes in Computer Science
We present an approximation algorithm for the problem that achieves a logarithmic approximation ratio by iteratively applying the algorithm for nding maximum weighted cliques.  ...  This problem is APX-hard for polygons without holes (i.e., there exists a constant > 0 such that no polynomial time algorithm can achieve an approximation ratio of 1 + ).  ...  As for computational complexity results, Minimum Convex Cover with(out) Holes can be approximated with a logarithmic approximation ratio 8].  ...

### Algorithms for connected set cover problem and fault-tolerant connected set cover problem

Zhao Zhang, Xiaofeng Gao, Weili Wu
2009 Theoretical Computer Science
These are the first approximation algorithms for CSC problems in general graphs with guaranteed performance ratios.  ...  In this paper, we present two approximation algorithms for the minimum CSC problem, and one approximation algorithm for the minimum (2, m)-CSC problem. Performance ratios are analyzed.  ...  For the general case, there is no known approximation algorithm with guaranteed performance ratio.  ...

### Optimal simplification of polygonal chains for subpixel-accurate rendering

Lilian Buzer
2009 Computational geometry
For a given polygonal chain, we study the min-# problem, which consists of finding an approximate and ordered subchain with a minimum number of vertices under a given approximation criterion.  ...  Previous algorithms with near-linear time performance produced geometrical inconsistencies and former methods used to preserve the features of the original chain required a quadratic time complexity.  ...  Acknowledgements The author would like to thank the reviewers for their helpful suggestions.  ...

### An optimal resource allocation with joint carrier aggregation in 4G-LTE

2015 2015 International Conference on Computing, Networking and Communications (ICNC)
In addition, every user subscribing for the mobile service is guaranteed to have a minimum quality-of-service (QoS) with a priority criterion.  ...  Finally, we present simulation results for the performance of our rate allocation algorithm.  ...  Therefore, the algorithm gives priority to the users with adaptive real-time applications with guaranteed minimum QoS for all service subscribers.  ...

### Approximation of Multiobjective Optimization Problems [chapter]

Mihalis Yannakakis
2001 Lecture Notes in Computer Science
We provide approximation algorithms with tight performance guarantees for bi-objective problems and make progress for the more challenging case of three and more objectives.  ...  We characterize when such an approximation can be efficiently constructed and investigate the problem of computing minimum size approximate convex Pareto sets, both for discrete and convex problems.  ...  Is there a generalization of the Chord algorithm with a poly-logarithmic performance guarantee? And what is the optimal ratio attainable by any algorithm with access to a Comb routine?  ...

### A logarithmic approximation for polymatroid congestion games

Tobias Harks, Tim Oosterwijk, Tjark Vredeveld
2016 Operations Research Letters
For general non-decreasing cost functions we devise an H rk -approximation algorithm, where rk is the sum of the ranks of the player-specific polymatroids and H rk denotes the rk-th harmonic number.  ...  The approximation guarantee is best possible up to a constant factor. As a special case, our result (partially) settles an open problem of Ackermann et al. (H. Ackermann, H. Röglin, and B. Vöcking.  ...  We thank Rico Zenklusen for his contribution to Lemma 2, Rudi Pendavingh and Jorn van der Pol for their contribution to the matroid version of this lemma, and an anonymous reviewer for his helpful comments  ...

### Page 635 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews
Summary: “Polynomial-time approximation algorithms with non- trivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph into k blocks so as to maximize  ...  It is shown that Nesterov and Nemirovskii’s universal barrier function of a convex cone is the logarithm of the convex cone’s characteris- tic function.  ...

### Logarithmic regret algorithms for online convex optimization

Elad Hazan, Amit Agarwal, Satyen Kale
2007 Machine Learning
In this paper, we give algorithms that achieve regret O(log(T )) for an arbitrary sequence of strictly convex functions (with bounded first and second derivatives).  ...  Zinkevich showed that a simple online gradient descent algorithm achieves additive regret O( √ T ), for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to the best  ...  Acknowledgements Many thanks to Adam Kalai for many critical contributions to this research. We would also like to thank Sanjeev Arora, Rob Schapire and an anonymous referee for helpful comments.  ...

### Rank minimization via online learning

Raghu Meka, Prateek Jain, Constantine Caramanis, Inderjit S. Dhillon
2008 Proceedings of the 25th international conference on Machine learning - ICML '08
A salient feature of our online learning approach is that it allows us to give provable approximation guarantees for the rank minimization problem over polyhedral sets.  ...  Minimum rank problems arise frequently in machine learning applications and are notoriously difficult to solve due to the non-convex nature of the rank objective.  ...  (Barvinok, 2002 ) (Chapter V) describes an approximation algorithm for RMP based on random projections and a generalization of the Johnson-Lindenstrauss Lemma, with an approximation guarantee similar to  ...

### Stackelberg Network Pricing Games [article]

Patrick Briest, Martin Hoefer, Piotr Krysta
2008 arXiv   pre-print
If followers have demands, the single-price algorithm provides a (1+ϵ)m^2-approximation, and the problem is hard to approximate within O(m^ϵ) for some ϵ >0.  ...  Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a (1+ϵ) m-approximation for any ϵ >0.  ...   present a first polynomial time approximation algorithm with a provable performance guarantee, which yields logarithmic approximation ratios. Bouhtou et al.  ...
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