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Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems

Joseph S. B. Mitchell
1999 SIAM journal on computing (Print)  
), and the k-MST problem.  ...  The method is based on the concept of an "m-guillotine subdivision," a simple extension of the recent approximation method of Mitchell [9] , which considered the case m = 1.  ...  I thank Esther Arkin, Rafi Hassin, Samir Khuller, Günter Rote, and the referees for several useful comments and suggestions that improved the presentation of the paper.  ... 
doi:10.1137/s0097539796309764 fatcat:qkugndrzgjanfonb3bwgrtdx5m

A Constant-Factor Approximation Algorithm for the Geometrick-MST Problem in the Plane

Joseph S. B. Mitchell, Avrim Blum, Prasad Chalasani, Santosh Vempala
1998 SIAM journal on computing (Print)  
We introduce a new technique that can be used to obtain simple approximation algorithms for geometric network design problems. The method is based on the concept of a \guillotine subdivision".  ...  To illustrate the power of the method, we show how it can be used to give a very simple constant-factor approximation algorithm for the geometric k-MST problem, obtaining a substantially better factor  ...  Mata for comments and suggestions that improved this paper.  ... 
doi:10.1137/s0097539796303299 fatcat:fibg35t3arh73dpft2roqat73q

Page 7914 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
B. (1-SUNYS-S; Stony Brook, NY) Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric kK-MST problem.  ...  In particular, a consequence of our main theorem is a very simple proof that the k-MST problem in the plane has a constant factor polynomial-time approximation al- , 909 ECONOMICS, OPERATIONS RESEARCH,  ... 

Page 4944 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
for the geometric k-MST problem in the plane.  ...  In particular, a consequence of our main theorem is a very simple proof that the A-MST problem in the plane has a constant-factor polynomial-time approximation algo- rithm: we obtain a factor of 2 [resp  ... 

Spanning Trees and Spanners [chapter]

David Eppstein
2000 Handbook of Computational Geometry  
We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.  ...  We describe the 2 √ 2 approximation algorithm, but the other results use similar methods. For related work on non-geometric k-MST problems see [13, 22, 30, 97, 116] .  ...  Up to constant factors in the approximation ratio, the k-MST problem is equivalent to the problem of finding a path connecting k points (the k-TSP problem) or a Steiner tree connecting k points.  ... 
doi:10.1016/b978-044482537-7/50010-3 fatcat:gitonpgfozgfribivszd6gf5cy

A constant-factor approximation algorithm for the k MST problem (extended abstract)

Avrim Blum, R. Ravi, Santosh Vempala
1996 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96  
In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points.  ...  We present the first constant-factor approximation algorithm for TSPN on an arbitrary set of disjoint, connected neighborhoods in the plane.  ...  We convert T into a polygonal subdivision, G, each of whose faces are histograms; this 2 increases the total length of T by only a constant factor. 4 .  ... 
doi:10.1145/237814.237992 dblp:conf/stoc/BlumRV96 fatcat:ubcvflvuu5c33g5j5kdp4t4ksu

9707-Computational Geometry

Rolf Klein, Hagen, Raimund Seidel
unpublished
The aim of this workshop was twofold: Firstly, to provide a forum for discussions about the interplay between the theory and practice of geometric computing.  ...  In particular it gave a chance to hear about (and maybe influence) the nascent efforts of various groups to create libraries of geometric algorithms.  ...  A Simple Polynomial-Type Approximation Scheme for Geometric Network Optimization Joe Mitchell We present a simple polynomial-time approximation scheme for geometric instances of the k-MST problem, the  ... 
fatcat:7m52stmf7vbwbosir6zcbjnosi

Page 1523 of Mathematical Reviews Vol. 28, Issue Index [page]

Mathematical Reviews  
Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric kK-MST problem.  ...  (English summary) 96a:68041 Bhavnagri, Burzin A method for representing shape based on an equivalence relation on polygons.  ... 

Page 1911 of Mathematical Reviews Vol. 32, Issue Index [page]

Mathematical Reviews  
Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems.  ...  (with Rao, Satish) A nearly linear-time approximation scheme for the Euclidean &-median problem.  ... 

Page 988 of Mathematical Reviews Vol. 28, Issue Index [page]

Mathematical Reviews  
Counting convex polygons in planar point sets. (English summary) 96h:68212 — Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric kK-MST problem.  ...  B.; et al., 96h:68212 Wu, Hui An optimal algorithm for finding the internal common tangent of two simple planar polygons. (Chinese.  ... 

TSP on manifolds [article]

David Zisselman
2021 arXiv   pre-print
Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial time approximation scheme for geometric TSP, k- MST, and related problems. SIAM J. Comput., 28(4):1298-1309, 1999.  ...  In this paper, we present a new approach of creating PTAS to the TSP problems by defining a bounded-curvature surface embedded spaces.  ...  Given a series of metric spaces d k : {1, . . . , n k } → R for k ∈ N one can estimate the 1 C approximation of the TSP problem on d k (with probability ≥ 1 2 ).  ... 
arXiv:2110.01093v1 fatcat:umnnrvmxyrccdka5yiyept3vdi

Page 1275 of Mathematical Reviews Vol. 32, Issue Index [page]

Mathematical Reviews  
(see 2000j:68006) 68U0S — Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems.  ...  Sweeping simple polygons with a chain of guards.  ...