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## Filters

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

1999
*
SIAM journal on computing (Print)
*

), and

doi:10.1137/s0097539796309764
fatcat:qkugndrzgjanfonb3bwgrtdx5m
*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. ...##
###
A Constant-Factor Approximation Algorithm for the Geometrick-MST Problem in the Plane

1998
*
SIAM journal on computing (Print)
*

We introduce

doi:10.1137/s0097539796303299
fatcat:fibg35t3arh73dpft2roqat73q
*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. ...##
###
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]

2000
*
Handbook of Computational Geometry
*

We survey results in

doi:10.1016/b978-044482537-7/50010-3
fatcat:gitonpgfozgfribivszd6gf5cy
*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. ...##
###
A constant-factor approximation algorithm for the k MST problem (extended abstract)

1996
*
Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
*

In

doi:10.1145/237814.237992
dblp:conf/stoc/BlumRV96
fatcat:ubcvflvuu5c33g5j5kdp4t4ksu
*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 . ...##
###
9707-Computational Geometry

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*...

##
###
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]

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 ). ...

##
###
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. ...