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Polynomial time approximation schemes for geometric k-clustering

R. Ostrovsky, Y. Rabani
Proceedings 41st Annual Symposium on Foundations of Computer Science  
We give polynomial time approximation schemes for this problem in several settings, including the binary cube f0; 1g d with Hamming distance, and R d either with L 1 distance, or with L 2 distance, or  ...  The problem is provably NP-hard in some high dimensional geometric settings, even for k = 2.  ...  1=d ) -time exact algorithm and a polynomial time approximation scheme with running time O(n logk) + (k= ) O(k 1?  ... 
doi:10.1109/sfcs.2000.892123 dblp:conf/focs/OstrovskyR00 fatcat:crozde7xvzhdzavwwy4vod6wlu

Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension

Michael Khachay, Katherine Neznakhina
2016 IFAC-PapersOnLine  
For the Euclidean Min-k-SCCP in R d , we construct a polynomial-time approximation scheme, which generalizes the approach proposed earlier for the planar Min-2-SCCP.  ...  For any fixed c > 1, the scheme finds Keywords: cycle cover of size k, traveling salesman problem (TSP), NP-hard problem, polynomial-time approximation scheme (PTAS).  ...  For instance, the Metric TSP (Christofides, 1975) can be approximated in polynomial time with a ratio of 3/2, and, for the Euclidean TSP, a polynomial-time approximation scheme (Arora, 1998 ) and a  ... 
doi:10.1016/j.ifacol.2016.07.541 fatcat:jzsopddbcfb4zjne2ij4lnpsha

Approximating minimum independent dominating sets in wireless networks

Johann L. Hurink, Tim Nieberg
2008 Information Processing Letters  
We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth.  ...  The presented approach yields a robust algorithm, that is, the algorithm accepts any undirected graph as input, and returns a (1 + ε)-approximate minimum dominating set, or a certificate showing that the  ...  We present a polynomial-time approximation scheme for this problem on graph with polynomially bounded growth.  ... 
doi:10.1016/j.ipl.2008.09.021 fatcat:2soqt3llgredjkkhtl5p57ds7a

Subject Index

2006 Journal of Discrete Algorithms  
Polar SAT Polar SAT and related graphs, 155 Polygonal line simplification Area-preserving approximations of polygonal paths, 554 Polynomial time Polynomial recognition of equal unions in hyper-  ...  schemes, 215 The layered subset difference (LSD) scheme On the mean number of encryptions for tree-based broadcast encryption schemes, 215 The subset difference (SD) scheme On the mean number of encryptions  ... 
doi:10.1016/s1570-8667(06)00094-3 fatcat:e5in7ez4svbdlcjyzzev4xkzhm

PRECISE

Shankar Krishnan, Mark Foskey, Tim Culver, John Keyser, Dinesh Manocha
2001 Proceedings of the seventeenth annual symposium on Computational geometry - SCG '01  
The algorithms designed for these problems make decisions based on signs of geometric predicates or on the roots of polynomials characterizing the problem.  ...  We also present a novel algorithm to compute all the roots of a univariate polynomial to any desired accuracy.  ...  Choice of Initial Approximations One of the critical steps for this iteration scheme to work is the choice of the initial approximations to the roots of the original polynomial.  ... 
doi:10.1145/378583.378693 dblp:conf/compgeom/KrishnanFCKM01 fatcat:j7plrx3j2rdbllp7yo5abnhsmi

The geometric generalized minimum spanning tree problem with grid clustering

Corinne Feremans, Alexander Grigoriev, René Sitters
2006 4OR  
We construct an exact exponential time dynamic programming algorithm and, based on this dynamic programming algorithm, we develop a polynomial time approximation scheme for the problem with grid clustering  ...  We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance defines the edge cost.  ...  Acknowledgments This research was supported by the Netherlands Organization for Scientific Research NWO, project "Treewidth and Combinatorial Optimization" (TACO).  ... 
doi:10.1007/s10288-006-0012-6 fatcat:w3n5riz5ynf2jowsgvlduiuqgu

Geometric Clustering to Minimize the Sum of Cluster Sizes [chapter]

Vittorio Bilò, Ioannis Caragiannis, Christos Kaklamanis, Panagiotis Kanellopoulos
2005 Lecture Notes in Computer Science  
For Euclidean spaces of higher dimensions, we show that the problem is NP-hard and present polynomial time approximation schemes.  ...  The latter result yields an improved approximation algorithm for the related problem of k-clustering to minimize the sum of cluster diameters.  ...  Our polynomial-time approximation scheme for min-size k-clustering with α = 1 yield a (2 + )-approximation algorithm for any constant dimension.  ... 
doi:10.1007/11561071_42 fatcat:lizrndztp5atrli5pihapzt7ra

Approximation of Geometric Dispersion Problems

C. Baur, S. P. Fekete
2001 Algorithmica  
While Hochbaum and Maass (1985) have given a polynomial time approximation scheme for packing, we show that geometric dispersion problems cannot be approximated arbitrarily well in polynomial time, unless  ...  We give a ~ approximation algorithm for one version of the geometric dispersion problem.  ...  motivation for this research.  ... 
doi:10.1007/s00453-001-0022-x fatcat:jna34zsncret5gh76fxb5w4hf4

Approximation of geometric dispersion problems [chapter]

Christoph Baur, Sándor P. Fekete
1998 Lecture Notes in Computer Science  
While Hochbaum and Maass (1985) have given a polynomial time approximation scheme for packing, we show that geometric dispersion problems cannot be approximated arbitrarily well in polynomial time, unless  ...  We give a ~ approximation algorithm for one version of the geometric dispersion problem.  ...  motivation for this research.  ... 
doi:10.1007/bfb0053964 fatcat:exwq7qjnoffxrnuxmjnjc7w4be

Approximation algorithms for Hamming clustering problems

Leszek Ga̧sieniec, Jesper Jansson, Andrzej Lingas
2004 Journal of Discrete Algorithms  
Sci. 38 (1985) 293-306], HRC and HDC can be approximated within a factor of two in time O(pkn). Next, we describe a 2 O(p /ε) k O(p/ε) n 2 -time (1 + ε)-approximation algorithm for HRC.  ...  In particular, it runs in polynomial time when p = O(1) and = O(log(k + n)).  ...  This was followed by a polynomial-time approximation scheme (PTAS) for 1-HRC [13] .  ... 
doi:10.1016/s1570-8667(03)00079-0 fatcat:xarko2kowzglna7due7uzhf744

Partition-Merge: Distributed Inference and Modularity Optimization [article]

Vincent Blondel, Kyomin Jung, Pushmeet Kohli, Devavrat Shah
2013 arXiv   pre-print
Here we say a graph has geometric structures, or polynomial growth property, when the number of nodes within distance r of any given node grows no faster than a polynomial function of r.  ...  We show that the resulting distributed algorithms for these problems essentially run in time linear in the number of nodes in the graph, and perform as well -- or even better -- than the original centralized  ...  To start with, Theorem 1(a) suggests that when graphs have polynomial growth, there exists a Randomized Polynomial Time Approximation Scheme (PTAS) for MAP computation and modularity optimization that  ... 
arXiv:1309.6129v1 fatcat:acmayfzg2na6dc6qvmcwe3f6mm

Approximation Algorithms for Hamming Clustering Problems [chapter]

Leszek Gąasieniec, Jesper Jansson, Andrzej Lingas
2000 Lecture Notes in Computer Science  
Sci. 38 (1985) 293-306], HRC and HDC can be approximated within a factor of two in time O(pkn). Next, we describe a 2 O(p /ε) k O(p/ε) n 2 -time (1 + ε)-approximation algorithm for HRC.  ...  In particular, it runs in polynomial time when p = O(1) and = O(log(k + n)).  ...  This was followed by a polynomial-time approximation scheme (PTAS) for 1-HRC [13] .  ... 
doi:10.1007/3-540-45123-4_11 fatcat:ibkv67l4sjcadpoutkohoju2ku

Page 7470 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
P. (3-WIND-FB; Windsor, ON) An efficient, strongly polynomial, ¢-approximation parametric optimization scheme. (English summary) Inform. Process. Lett. 64 (1997), no. 4, 173-177.  ...  An enumerating algo- rithm is called polynomial delay if every additional extreme point is Output in polynomial time and polynomial space of the input size.  ... 

Page 4202 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews  
Next, we describe a 22'7¢/©)k °'?/")n?-time (1 +e€)-approximation algorithm for HRC. In particular, it runs in polynomial time when p = O(1) and g = O(log(k + n)).  ...  On the other hand we have a polynomial-time approximation scheme for the Euclidean TSP and the }-approximation algorithm of Christofides for TSP instances satisfying the triangle inequal- ity.  ... 

Geometrical versus time-series representation of data in learning quantum control [article]

M. Ostaszewski, J.A. Miszczak, P. Sadowski
2018 arXiv   pre-print
We study the application of machine learning methods based on a geometrical and time-series character of data in the application to quantum control.  ...  We also show that the utilisation of the geometrical structure of control pulses is sufficient for achieving high-fidelity in quantum control using machine learning procedures.  ...  Clustering/classification algorithm The considered geometric correction scheme approximation method is based on two steps.  ... 
arXiv:1803.05169v1 fatcat:vjhquxbfjzblpgiv3md3ta7hc4
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