## Guillotine subdivisions approximate polygonal subdivisions: Part II - A simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problems (1996)

### Cached

### Download Links

- [ams.sunysb.edu]
- [ftp.ams.sunysb.edu]
- [ftp.ams.sunysb.edu]
- DBLP

### Other Repositories/Bibliography

Citations: | 163 - 12 self |

### BibTeX

@MISC{Mitchell96guillotinesubdivisions,

author = {Joseph S. B. Mitchell},

title = {Guillotine subdivisions approximate polygonal subdivisions: Part II - A simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problems},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper, thereby achieving essentially the same results that we report here, using decomposition schemes that are somewhat similar to our own. Arora's remarkable results predate this paper by several weeks, and his discovery was done independently of this work. 2 m-Guillotine Subdivisions

### Citations

9279 | Introduction to Algorithms
- Cormen, Leiserson, et al.
- 2009
(Show Context)
Citation Context ...h of research on the problems studied here, both in the graph versions of the problems and in the geometric versions. Almost any standard textbook on algorithms and networks discusses them; e.g., see =-=[4, 6, 12]-=-. For a survey of work on the TSP, refer to the # Received by the editors September 25, 1996; accepted for publication (in revised form) July 18, 1997; published electronically March 22, 1999. This re... |

1894 |
Computational Geometry { an Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...um total length that spans (visits) the set of points P . In this problem, as opposed to the minimum spanning tree (MST) problem (solvable exactly in O(n log n) time for points in the Euclidean plane =-=[13]), the tre-=-e is allowed to have vertices ("Steiner points") that are not among the points of P . Steiner k-MST problem: For a given integer k (k # n), determine a tree, possibly with Steiner points, of... |

1394 |
Combinatorial Optimization: Algorithms and Complexity
- Papadimitriou, Steiglitz
- 1982
(Show Context)
Citation Context ...h of research on the problems studied here, both in the graph versions of the problems and in the geometric versions. Almost any standard textbook on algorithms and networks discusses them; e.g., see =-=[4, 6, 12]-=-. For a survey of work on the TSP, refer to the # Received by the editors September 25, 1996; accepted for publication (in revised form) July 18, 1997; published electronically March 22, 1999. This re... |

503 |
Combinatorial Optimization: Networks and Matroids
- Lawler
- 1976
(Show Context)
Citation Context ...h of research on the problems studied here, both in the graph versions of the problems and in the geometric versions. Almost any standard textbook on algorithms and networks discusses them; e.g., see =-=[4, 6, 12]-=-. For a survey of work on the TSP, refer to the # Received by the editors September 25, 1996; accepted for publication (in revised form) July 18, 1997; published electronically March 22, 1999. This re... |

345 | Polynomial time approximation schemes for Euclidean TSP and other geometric problems, Proc. 37th Annu
- Arora
- 1996
(Show Context)
Citation Context ...ification to a previous result [9, 11] (see also [3]) leads to a PTAS for various geometric optimization problems, including the TSP, Steiner tree, and k-MST. In an exciting recent development, Arora =-=[1]-=- announced that he had found a PTAS for the Euclidean TSP, as well as the other problems considered in this paper, thereby achieving essentially the same results that we report here, using decompositi... |

334 |
The Traveling Salesman Problem
- Lawler, Lenstra, et al.
- 1983
(Show Context)
Citation Context ...omp/28-4/30976.html + Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600 (jsbm@ams.sunysb.edu). A SIMPLE PTAS FOR GEOMETRIC TSP, ETC. 1299 book =-=[7]-=- edited by Lawler et al. For a survey on approximation algorithms, refer to the recent book [5] edited by Hochbaum. All of the geometric optimization problems considered here are known to be NPhard. P... |

292 | Combinatorial Optimization: Networks and - Lawler - 1976 |

91 | Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
- Arora
- 1998
(Show Context)
Citation Context .... We are currently examining possible extensions to higher dimensions. Finally, we mention some recent results that have been discovered since the time that this paper was originally submitted. Arora =-=[2]-=- has recently obtained a randomized A SIMPLE PTAS FOR GEOMETRIC TSP, ETC. 1309 algorithm with expected running time that is nearly linear in n: O(n log O(1/#) n). His new results also allow the d-dime... |

64 | Approximation Algorithms for Geometric Tour and Network Design Problems - S, MITCHELL |

31 | A constant-factor approximation for the k-MST problem in the plane
- Blum, Chalasani, et al.
- 1995
(Show Context)
Citation Context ... known. In particular, no factor better than the Christofides bound of 1.5 was known for the Euclidean TSP. In this paper, we point out how a minor modification to a previous result [9, 11] (see also =-=[3]-=-) leads to a PTAS for various geometric optimization problems, including the TSP, Steiner tree, and k-MST. In an exciting recent development, Arora [1] announced that he had found a PTAS for the Eucli... |

10 |
Approximation algorithms for geometric optimization problems
- Mitchell
(Show Context)
Citation Context ... nearly linear in n: O(n log O(1/#) n). His new results also allow the d-dimensional problem to be solved in expected time O(n(log n) (O( d # )) d-1 ). Further, in a follow-up to this paper, Mitchell =-=[10] has shown-=- how a variant of m-guillotine subdivisions (termed "grid-rounded m-guillotine subdivisions") yields a (deterministic) PTAS with worst-case time bound n O(1) , for any fixed # > 0, for the s... |

3 |
A constant-factor approximation for the geometric k-MST problem in the plane
- Mitchell, Blum, et al.
- 1999
(Show Context)
Citation Context ... scheme (PTAS) was known. In particular, no factor better than the Christofides bound of 1.5 was known for the Euclidean TSP. In this paper, we point out how a minor modification to a previous result =-=[9, 11]-=- (see also [3]) leads to a PTAS for various geometric optimization problems, including the TSP, Steiner tree, and k-MST. In an exciting recent development, Arora [1] announced that he had found a PTAS... |

2 |
More efficient approximation schemes for Euclidean TSP and other geometric problems, Unpublished manuscript
- Arora
- 1997
(Show Context)
Citation Context .... We are currently examining possible extensions to higher dimensions. Finally, we mention some recent results that have been discovered since the time that this paper was originally submitted. Arora =-=[2]-=- has recently obtained a randomized algorithm with expected running time that is nearly linear in n: O(n log O(1=ffl) n). His new results also allow the d-dimensional problem to be solved in expected ... |