Augmenting Outerplanar Graphs to Meet Diameter Requirements
Given an undirected graph G=(V,E) and an integer D≥1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP, whil...
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| Published in: | Journal of graph theory Vol. 74; no. 4; pp. 392 - 416 |
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| Format: | Journal Article |
| Language: | English |
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| ISSN: | 0364-9024, 1097-0118 |
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| Abstract | Given an undirected graph G=(V,E) and an integer D≥1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP, while the problem with only a few graph classes such as forests is approximable within a constant factor. In this article, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G. We also show that if the target diameter D is even, then the case where G is a partial 2‐tree is also approximable within a constant. |
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| AbstractList | Given an undirected graph G=(V,E) and an integer D≥1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP, while the problem with only a few graph classes such as forests is approximable within a constant factor. In this article, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G. We also show that if the target diameter D is even, then the case where G is a partial 2‐tree is also approximable within a constant. Given an undirected graph and an integer , we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D . It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless , while the problem with only a few graph classes such as forests is approximable within a constant factor. In this article, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G . We also show that if the target diameter D is even, then the case where G is a partial 2‐tree is also approximable within a constant. Given an undirected graph G = ( V , E ) and an integer D ≥ 1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless P = N P, while the problem with only a few graph classes such as forests is approximable within a constant factor. In this article, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G. We also show that if the target diameter D is even, then the case where G is a partial 2-tree is also approximable within a constant. [PUBLICATION ABSTRACT] |
| Author | Ishii, Toshimasa |
| Author_xml | – sequence: 1 givenname: Toshimasa surname: Ishii fullname: Ishii, Toshimasa email: ishii@res.otaru-uc.ac.jp organization: DEPARTMENT OF INFORMATION AND MANAGEMENT SCIENCE, OTARU UNIVERSITY OF COMMERCE, 047-8501, OTARU, JAPAN |
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| Cites_doi | 10.1137/S0895480193253415 10.1007/s004930050035 10.1016/0196-6774(86)90023-4 10.1002/jgt.3190110315 10.1002/1097-0118(200011)35:3<161::AID-JGT1>3.0.CO;2-Y 10.1016/j.tcs.2011.05.014 10.1145/301250.301447 10.1016/0167-6377(92)90007-P 10.1007/3-540-45477-2_19 10.1007/s00453-005-1183-9 10.1016/j.disopt.2005.10.006 10.1007/978-3-540-74208-1_5 10.1007/s00453-001-0113-8 10.1145/1077464.1077468 10.1137/S0097539793251219 |
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| Copyright | 2013 Wiley Periodicals, Inc. Copyright © 2013 Wiley Periodicals, Inc. |
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| Notes | istex:6D44827327797D04192FB30DA0828FE8BEC8CCC1 ark:/67375/WNG-198Q6NQ2-K ArticleID:JGT21719 An extended abstract of this article was presented at 18th Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia, February 2012, pp. 123-132. Ministry of Education, Culture, Sports, Science and Technology of Japan Kayamori Foundation of Informational Science Advancement An extended abstract of this article was presented at 18th Computing: The Australasian Theory Symposium (CATS 2012), Melbourne, Australia, February 2012, pp. 123–132. Contract grant sponsor: Ministry of Education, Culture, Sports, Science and Technology of Japan; Contract grant sponsor: Kayamori Foundation of Informational Science Advancement. ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
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| References_xml | – reference: H. L. Bodlaender, a linear time algorithm for finding tree-decompositions of small treewidth, SIAM J Comput 25 (1996), 1305-1317. – reference: N. Alon, A. Gyárfás, and M. Ruszinkó, Decreasing the diameter of bounded degree graphs, J Graph Theory 35 (2000), 161-172. – reference: C. Berge, Hypergraphs, Elsevier, North-Holland, Amsterdam, 1989. – reference: A. Brandstädt, F. Dragan, V. Chepoi, and V. Voloshin, Dually chordal graphs, SIAM J Discrete Math 11(3) (1998), 437-455. – reference: N. Robertson and P. D. Seymour, Graph minors. II. Algorithmic aspects of tree-width, J Algorithms 7 (1986), 309-322. – reference: V. Chepoi and Y. Vaxès, Augmenting trees to meet biconnectivity and diameter constraints, Algorithmica 33 (2002), 243-262. – reference: E. D. Demaine, F. V. Fomin, M. Hajiaghayi, and D. M. Thilikos, Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs, ACM Trans Algorithms 1 (2005), 33-47. – reference: P. Erdős, A. Gyárfás, and M. Ruszinkó, How to decrease the diameter of triangle-free graphs, Combinatorica 18(4) (1998), 493-501. – reference: T. Ishii, S. Yamamoto, and H. Nagamochi, Augmenting forests to meet odd diameter requirements, Discrete Optim 3 (2006), 154-164. – reference: D. Bilò, L. Gualà, and G. Proietti, Improved approximability and non-approximability results for graph diameter decreasing problems, Theor Comput Sci 417 (2012), 12-22. – reference: V. Chepoi, B. Estellon, K. Nouioua, and Y. Vaxès, Mixed covering of trees and the augmentation problem with odd diameter constraints, Algorithmica 45 (2006), 209-226. – reference: A. A. Schoone, H. L. Bodlaendar, and J. van Leeuwen, Diameter increase caused by edge deletion, J Graph Theory 11 (3) (1987), 409-427. – reference: F. R. K. Chung, Diameter of graphs: Old problems and new results, Congr Numer 60 (1987), 295-317. – reference: C. Li, S. McCormick, and D. Simchi-Levi, On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems, Oper Res Lett 11 (1992), 303-308. – volume: 25 start-page: 1305 year: 1996 end-page: 1317 article-title: a linear time algorithm for finding tree‐decompositions of small treewidth publication-title: SIAM J Comput – article-title: Algorithmes de couverture et d'augmentation de graphes sous contraintes de distance – volume: 417 start-page: 12 year: 2012 end-page: 22 article-title: Improved approximability and non‐approximability results for graph diameter decreasing problems publication-title: Theor Comput Sci – volume: 33 start-page: 243 year: 2002 end-page: 262 article-title: Augmenting trees to meet biconnectivity and diameter constraints publication-title: Algorithmica – start-page: 201 year: 2001 end-page: 216 article-title: Small ‐dominating sets in planar graphs with applications – volume: 35 start-page: 161 year: 2000 end-page: 172 article-title: Decreasing the diameter of bounded degree graphs publication-title: J Graph Theory – volume: 60 start-page: 295 year: 1987 end-page: 317 article-title: Diameter of graphs: Old problems and new results publication-title: Congr Numer – volume: 7 start-page: 309 year: 1986 end-page: 322 article-title: Graph minors. II. Algorithmic aspects of tree‐width publication-title: J Algorithms – year: 1989 – volume: 1 start-page: 33 year: 2005 end-page: 47 article-title: Fixed‐parameter algorithms for ( )‐center in planar graphs and map graphs publication-title: ACM Trans Algorithms – volume: 3 start-page: 154 year: 2006 end-page: 164 article-title: Augmenting forests to meet odd diameter requirements publication-title: Discrete Optim – start-page: 750 year: 1999 end-page: 759 article-title: Designing networks with bounded pairwise distance – volume: 11 start-page: 437 issue: 3 year: 1998 end-page: 455 article-title: Dually chordal graphs publication-title: SIAM J Discrete Math – volume: 45 start-page: 209 year: 2006 end-page: 226 article-title: Mixed covering of trees and the augmentation problem with odd diameter constraints publication-title: Algorithmica – volume: 18 start-page: 493 issue: 4 year: 1998 end-page: 501 article-title: How to decrease the diameter of triangle‐free graphs publication-title: Combinatorica – volume: 11 start-page: 303 year: 1992 end-page: 308 article-title: On the minimum‐cardinality‐bounded‐diameter and the bounded‐cardinality‐minimum‐diameter edge addition problems publication-title: Oper Res Lett – article-title: Packing and covering δ‐hyperbolic spaces by balls, approximation, randomization, and combinatorial optimization – volume: 11 start-page: 409 issue: 3 year: 1987 end-page: 427 article-title: Diameter increase caused by edge deletion publication-title: J Graph Theory – ident: e_1_2_8_6_1 doi: 10.1137/S0895480193253415 – ident: e_1_2_8_13_1 doi: 10.1007/s004930050035 – ident: e_1_2_8_14_1 – ident: e_1_2_8_18_1 doi: 10.1016/0196-6774(86)90023-4 – ident: e_1_2_8_19_1 doi: 10.1002/jgt.3190110315 – ident: e_1_2_8_2_1 doi: 10.1002/1097-0118(200011)35:3<161::AID-JGT1>3.0.CO;2-Y – ident: e_1_2_8_4_1 doi: 10.1016/j.tcs.2011.05.014 – ident: e_1_2_8_12_1 doi: 10.1145/301250.301447 – ident: e_1_2_8_17_1 doi: 10.1016/0167-6377(92)90007-P – ident: e_1_2_8_15_1 doi: 10.1007/3-540-45477-2_19 – volume: 60 start-page: 295 year: 1987 ident: e_1_2_8_10_1 article-title: Diameter of graphs: Old problems and new results publication-title: Congr Numer – ident: e_1_2_8_8_1 doi: 10.1007/s00453-005-1183-9 – ident: e_1_2_8_16_1 doi: 10.1016/j.disopt.2005.10.006 – ident: e_1_2_8_7_1 doi: 10.1007/978-3-540-74208-1_5 – volume-title: Hypergraphs year: 1989 ident: e_1_2_8_3_1 – ident: e_1_2_8_9_1 doi: 10.1007/s00453-001-0113-8 – ident: e_1_2_8_11_1 doi: 10.1145/1077464.1077468 – ident: e_1_2_8_5_1 doi: 10.1137/S0097539793251219 |
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| Snippet | Given an undirected graph G=(V,E) and an integer D≥1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at... Given an undirected graph and an integer , we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D . It is... Given an undirected graph G = ( V , E ) and an integer D ≥ 1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter... |
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| SubjectTerms | Algorithms constant factor approximation algorithm Decision trees diameter graph augmentation problem outerplanar graphs partial 2-trees undirected graph |
| Title | Augmenting Outerplanar Graphs to Meet Diameter Requirements |
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