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

André Berger, Artur Czumaj, Michelangelo Grigni, Hairong Zhao
2005 Lecture Notes in Computer Science  
From this we derive quasi-polynomial time 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.  ... 
doi:10.1007/11561071_43 fatcat:m565l4ypivaafiuyt6vvzq5goe

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Philip N. Klein
2008 SIAM journal on computing (Print)  
We give an algorithm requiring O(c 1/ǫ 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.  ... 
doi:10.1137/060649562 fatcat:qkvmtagkdvgrbobjn2zi2fgsku

The Two-Edge Connectivity Survivable Network Problem in Planar Graphs [chapter]

Glencora Borradaile, Philip Klein
2008 Lecture Notes in Computer Science  
Consider the following problem: given a 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.  ... 
doi:10.1007/978-3-540-70575-8_40 fatcat:krfe44tyjna25e4cpasvhm2cma

Approximate TSP in Graphs with Forbidden Minors [chapter]

Michelangelo Grigni
2000 Lecture Notes in Computer Science  
These results imply (i) a QPTAS (quasi-polynomial time 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  ... 
doi:10.1007/3-540-45022-x_73 fatcat:s3un4gipwfevhfyho2rfho4iiy

Object location using path separators

Ittai Abraham, Cyril Gavoille
2006 Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing - PODC '06  
We then show that k-path separable 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.  ... 
doi:10.1145/1146381.1146411 dblp:conf/podc/AbrahamG06 fatcat:t66mry554jfnvj6mhzeolvzhwu

A subset spanner for Planar graphs,

Philip N. Klein
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.  ... 
doi:10.1145/1132516.1132620 dblp:conf/stoc/Klein06 fatcat:2cjsn6tvx5fsbj7dhc33lklp4i

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

Prabhakar Gubbala, Balaji Raghavachari
2004 Lecture Notes in Computer Science  
O(logk) 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.  ... 
doi:10.1007/978-3-540-24698-5_25 fatcat:ialvj3bv6vegfhey6s7vjfld3a

Approximation of minimum weight spanners for sparse graphs

Feodor F. Dragan, Fedor V. Fomin, Petr A. Golovach
2011 Theoretical Computer Science  
A t-spanner of a 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.  ... 
doi:10.1016/j.tcs.2010.11.034 fatcat:r24q6gfu5be3djl7rwqhiyqfwy

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: // 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]

Baigong Zheng
2017 arXiv   pre-print
three-vertex 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.  ... 
arXiv:1701.08315v1 fatcat:zlw77dafundctdnctzkdktlyhi

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]

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.  ...  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.  ... 
doi:10.1016/b978-044482537-7/50010-3 fatcat:gitonpgfozgfribivszd6gf5cy

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

Jeff Erickson, Philip N. Klein, Dániel Marx, Claire Mathieu, Marc Herbstritt
2016 Dagstuhl Reports  
This report contains abstracts 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  ... 
doi:10.4230/dagrep.6.5.94 dblp:journals/dagstuhl-reports/EricksonKMM16 fatcat:wasdfgivt5fqdppfxo3iqqs2ta

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

Philip N. Klein, Claire Mathieu, Hang Zhou, Marc Herbstritt
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.  ... 
doi:10.4230/lipics.stacs.2015.554 dblp:conf/stacs/KleinMZ15 fatcat:7h7w2543qbdbxixjf25ijziacq
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