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Approximation Schemes for Minimum 2-Connected Spanning Subgraphs in Weighted Planar Graphs
[chapter]

2005
*
Lecture Notes in Computer Science
*

From this we derive quasi-polynomial time

doi:10.1007/11561071_43
fatcat:m565l4ypivaafiuyt6vvzq5goe
*approximation**schemes**for*the problems of finding the*minimum*-*weight**2*-edge-*connected*or biconnected*spanning**subgraph**in**planar**graphs*. ... We present new*approximation**schemes**for*various classical problems of finding the*minimum*-*weight**spanning**subgraph**in*edge-*weighted*undirected*planar**graphs*that are resistant to edge or vertex removal ... Let ε > 0, and let G be a*weighted**planar**graph*with n vertices. There is an algorithm running*in*time n O(log n·log(1/ε)/ε) that outputs a {1,2}-VCSS H of G such that w(H) ≤ (1 + ε) · OPT. ...##
###
A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

2008
*
SIAM journal on computing (Print)
*

We give an algorithm requiring O(c 1/ǫ

doi:10.1137/060649562
fatcat:qkvmtagkdvgrbobjn2zi2fgsku
*2*n) time to find an ǫ-optimal traveling salesman tour*in*the shortest-path metric defined by an undirected*planar**graph*with nonnegative edgelengths. ...*For*the case of all lengths equal to 1, the time required is O(c 1/ǫ n). ... Thanks also to Glencora Borradaile and Erik Demaine*for*helpful discussions and suggestions. ...##
###
The Two-Edge Connectivity Survivable Network Problem in Planar Graphs
[chapter]

2008
*
Lecture Notes in Computer Science
*

Consider the following problem: given a

doi:10.1007/978-3-540-70575-8_40
fatcat:krfe44tyjna25e4cpasvhm2cma
*graph*with edgeweights and a subset Q of vertices, find a*minimum*-*weight**subgraph**in*which there are two edge-disjoint paths*connecting*every pair of vertices*in*... The problem is SNP-hard*in*general*graphs*and NP-hard*in**planar**graphs*. We give the first polynomial-time*approximation**scheme**in**planar**graphs*. The running time is O(n log n). ...*In*[3] , Berger and Grigni gave a polynomial-time*approximation**scheme*(PTAS)*for*{1,*2*}-edge*connectivity*(ie. the*spanning*case)*in**planar*multigraphs. ...##
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Approximate TSP in Graphs with Forbidden Minors
[chapter]

2000
*
Lecture Notes in Computer Science
*

These results imply (i) a QPTAS (quasi-polynomial time

doi:10.1007/3-540-45022-x_73
fatcat:s3un4gipwfevhfyho2rfho4iiy
*approximation**scheme*)*for*the TSP (traveling salesperson problem)*in*unweighted*graphs*with an excluded minor, and (ii) a QPTAS*for*the TSP*in**weighted*... Given as input an edge-*weighted**graph*, we analyze two algorithms*for*nding*subgraphs*with low total edge*weight*. ... Acknowledgment: We thank Robin Thomas*for*help with Lemma 10. References ...##
###
Object location using path separators

2006
*
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing - PODC '06
*

We then show that k-path separable

doi:10.1145/1146381.1146411
dblp:conf/podc/AbrahamG06
fatcat:t66mry554jfnvj6mhzeolvzhwu
*graphs*can be used to solve several object location problems: (1) a small-worldization with an average poly-logarithmic number of hops; (*2*) an (1 + ε)*approximate*distance ... Our main result is that every minor free*weighted**graph*is k-path separable. ... This*weighting**scheme*generalizes the regular vertex*weighting*to capture the*connectivity*between the center*subgraph*and the remaining of the*graph*. ...##
###
A subset spanner for Planar graphs,

2006
*
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06
*

*For*any edge-

*weighted*

*planar*

*graph*G and a subset S of nodes of G, there is a

*subgraph*H of G of

*weight*a constant times that of the

*minimum*Steiner tree

*for*S such that distances

*in*H between nodes

*in*... As a consequence, there is an O(n log n)-time

*approximation*

*scheme*

*for*finding a TSP among a given subset of nodes of a

*planar*

*graph*. This is the first PTAS

*for*the problem. ... Acknowlegements Thanks to Glencora Borradaile and Claire Kenyon

*for*helpful discussions. ...

##
###
Finding k-Connected Subgraphs with Minimum Average Weight
[chapter]

2004
*
Lecture Notes in Computer Science
*

O(logk)

doi:10.1007/978-3-540-24698-5_25
fatcat:ialvj3bv6vegfhey6s7vjfld3a
*approximation**for*k-node-*connectivity*3.*2*+*approximation**for*k-node-*connectivity**in*Euclidian*graphs*,*for*any constant > 0, 4. 5.8-*approximation**for*k-node-*connectivity**in**graphs*satisfying the ... We consider the problems of finding k-*connected**spanning**subgraphs*with*minimum*average*weight*. We show that the problems are NP-hard*for*k > 1. ... We would like to thank Si Qing Zheng*for*posing Avg-kEc. This research was supported*in*part by the National Science Foundation under grant CCR-9820902. ...##
###
Approximation of minimum weight spanners for sparse graphs

2011
*
Theoretical Computer Science
*

A t-spanner of a

doi:10.1016/j.tcs.2010.11.034
fatcat:r24q6gfu5be3djl7rwqhiyqfwy
*graph*G is its*spanning**subgraph*S such that the distance between every pair of vertices*in*S is at most t times their distance*in*G. ...*For*t ≥ 5, the problem remains NP-hard*for**planar**graphs*and the*approximability*status of the problem on*planar**graphs*was open. ... Acknowledgements We would like to thank the anonymous referees*for*comments which improved the presentation of this paper. ...##
###
Page 1329 of Mathematical Reviews Vol. , Issue 2003B
[page]

2003
*
Mathematical Reviews
*

Summary: “We study some versions of the problem of finding the

*minimum*size*2*-*connected**subgraph*. This problem is NP- hard (even on cubic*planar**graphs*) and MAX SNP-hard. ... We show that the*minimum**2*-edge*connected**subgraph*problem can*For*the web version of Mathematical Reviews, see http: //www.ams.org/mathscinet 2003b:68223 be*approximated*to within 4—e*for*general*graphs*...##
###
Linear-time approximation schemes for planar minimum three-edge connected and three-vertex connected spanning subgraphs
[article]

2017
*
arXiv
*
pre-print

three-vertex

arXiv:1701.08315v1
fatcat:zlw77dafundctdnctzkdktlyhi
*connected**spanning**subgraph*problem*in*undirected*planar**graphs*. ... We present the first polynomial-time*approximation**schemes*, i.e., (1 + ϵ)-*approximation*algorithm*for*any constant ϵ > 0,*for*the*minimum*three-edge*connected**spanning**subgraph*problem and the*minimum*... Acknowledgements We thank Glencora Borradaile and Hung Le*for*helpful discussions. ...##
###
Page 5330 of Mathematical Reviews Vol. , Issue 2000h
[page]

2000
*
Mathematical Reviews
*

Summary: “Given a complete undirected

*graph*with nonnegative costs on the edges, the*2*-edge*connected**subgraph*problem consists*in*finding the*minimum*cost*spanning**2*-edge*connected**subgraph*(where multiedges ... Ravi, R. (1-CMU-I; Pittsburgh, PA) A new bound*for*the*2*-edge*connected**subgraph*problem. ...##
###
Spanning Trees and Spanners
[chapter]

2000
*
Handbook of Computational Geometry
*

We survey results

doi:10.1016/b978-044482537-7/50010-3
fatcat:gitonpgfozgfribivszd6gf5cy
*in*geometric network design theory, including algorithms*for*constructing*minimum**spanning*trees and low-dilation*graphs*. ... Very recently, Arora (personal communication) has discovered a polynomial time*approximation**scheme**for*the*planar*Traveling Salesman Problem. ... Clearly the*weight*should be measured*in*terms of the*minimum**spanning*tree,*for*as Das and Joseph [39] observe, any*graph*with bounded dilation should at least be*connected*. ...##
###
Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

2016
*
Dagstuhl Reports
*

This report contains abstracts

doi:10.4230/dagrep.6.5.94
dblp:journals/dagstuhl-reports/EricksonKMM16
fatcat:wasdfgivt5fqdppfxo3iqqs2ta
*for*the recent developments*in**planar**graph*algorithms discussed during the seminar as well as summaries of open problems*in*this area of research. ... This report documents the program and the outcomes of Dagstuhl Seminar 16221 "Algorithms*for*Optimization Problems*in**Planar**Graphs*". The seminar was held from May 29 to June 3, 2016. ... With that framework, we derive polynomial-time*approximation**schemes**for*the following problems*in**planar**graphs*or*graphs*of bounded genus: edge-*weighted*tree cover and tour cover; vertex-*weighted**connected*...##
###
Page 8131 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

“We also study hardness of

*approximations**for*the*minimum*- cost k-vertex- and k-edge-*connected**spanning**subgraph*problems. ... Summary: “We present the first truly polynomial-time approxima- tion*scheme*(PTAS)*for*the*minimum*-cost k-vertex- (or k-edge-)*connected**spanning**subgraph*problem*for*complete Euclidean*graphs**in*R°. ...##
###
Correlation Clustering and Two-edge-connected Augmentation for Planar Graphs

2015
*
Symposium on Theoretical Aspects of Computer Science
*

*For*

*planar*

*graphs*, we prove that correlation clustering reduces to two-edge-

*connected*augmentation, and that both problems have a polynomial-time

*approximation*

*scheme*. ... The goal is to produce a

*minimum*

*weight*subset S of edges of the

*graph*, such that

*for*every edge

*in*R, its endpoints are two-edge-

*connected*

*in*R ∪ S. ... Karloff

*for*helping make the

*connection*between correlation clustering and two-edge-

*connected*augmentation

*in*

*planar*

*graphs*; Nabil Mustafa

*for*numerous discussions; and Grigory Yaroslavtsev. ...

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