Document Open Access Logo

Obstructions to Faster Diameter Computation: Asteroidal Sets

Author Guillaume Ducoffe

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2022.10.pdf
  • Filesize: 0.84 MB
  • 24 pages

Document Identifiers

Author Details

Guillaume Ducoffe
  • National Institute of Research and Development in Informatics, Bucharest, Romania
  • University of Bucharest, Romania

Funding

This work was supported by project PN-19-37-04-01 “New solutions for complex problems in current ICT research fields based on modelling and optimization”, funded by the Romanian Core Program of the Ministry of Research and Innovation (MCI) 2019-2022. This work was also supported by a grant of the Ministry of Research, Innovation and Digitalization, CCCDI - UEFISCDI, project number PN-III-P2-2.1-PED-2021-2142, within PNCDI III.

Cite As Get BibTex

Guillaume Ducoffe. Obstructions to Faster Diameter Computation: Asteroidal Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.IPEC.2022.10

Abstract

An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let Ext_α be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than α pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every m-edge graph in Ext_{α} can be computed in deterministic O(α³ m^{3/2}) time. We then improve the runtime to O(α³m) for bipartite graphs, to O(α⁵m) for triangle-free graphs, O(α³Δm) for graphs with maximum degree Δ, and more generally to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive +1-approximation of all vertex eccentricities in deterministic O(α² m) time. This is in sharp contrast with general m-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in O(m^{2-ε}) time for any ε > 0.
As important special cases of our main result, we derive an O(m^{3/2})-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an O(k³m^{3/2})-time algorithm for this problem on graphs of asteroidal number at most k. Both results extend prior works on exact and approximate diameter computation within AT-free graphs. To the best of our knowledge, this is also the first deterministic subquadratic-time algorithm for computing the diameter within the subclasses of: chordal graphs of bounded leafage (generalizing the interval graphs), k-moplex graphs and k-polygon graphs (generalizing the permutation graphs) for any fixed k. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.
Our approach is purely combinatorial, that differs from most prior recent works in this area which have relied on geometric primitives such as Voronoi diagrams or range queries. On our way, we uncover interesting connections between the diameter problem, Lexicographic Breadth-First Search, graph extremities and the asteroidal number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Diameter computation
  • Asteroidal number
  • LexBFS

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. A. Abboud, V. Vassilevska Williams, and J. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the Twenty-seventh Annual ACM-SIAM symposium on Discrete Algorithms (SODA), pages 377-391. SIAM, 2016. Google Scholar
  2. J. Alman and V. Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 522-539. SIAM, 2021. Google Scholar
  3. H.-J. Bandelt, A. Dählmann, and H. Schütte. Absolute retracts of bipartite graphs. Discrete Applied Mathematics, 16(3):191-215, 1987. Google Scholar
  4. H.-J. Bandelt and E. Pesch. Efficient characterizations of n-chromatic absolute retracts. Journal of Combinatorial Theory, Series B, 53(1):5-31, 1991. Google Scholar
  5. J. Beisegel. Characterising AT-free graphs with BFS. In International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 15-26. Springer, 2018. Google Scholar
  6. A. Berry and J.-P. Bordat. Separability generalizes Dirac’s theorem. Discrete Applied Mathematics, 84(1-3):43-53, 1998. Google Scholar
  7. A. Berry and J.-P. Bordat. Local LexBFS properties in an arbitrary graph. In Proceedings of Journéees Informatiques Messines (JIM 2000), 2000. Google Scholar
  8. A. Berry and J.-P. Bordat. Asteroidal triples of moplexes. Discrete Applied Mathematics, 111(3):219-229, 2001. Google Scholar
  9. J. A. Bondy and U. S. R. Murty. Graph theory. Springer London, 2008. Google Scholar
  10. M. Borassi, P. Crescenzi, and L. Trevisan. An axiomatic and an average-case analysis of algorithms and heuristics for metric properties of graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 920-939. SIAM, 2017. Google Scholar
  11. N. Bousquet and S. Thomassé. VC-dimension and Erdős-Pósa property. Discrete Mathematics, 338(12):2302-2317, 2015. Google Scholar
  12. A. Brandstädt, V. Chepoi, and F.F. Dragan. The algorithmic use of hypertree structure and maximum neighbourhood orderings. Discrete Applied Mathematics, 82(1-3):43-77, 1998. Google Scholar
  13. A. Brandstädt, P. Fičur, A. Leitert, and M. Milanič. Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs. Information Processing Letters, 115(2):256-262, 2015. Google Scholar
  14. K. Bringmann, T. Husfeldt, and M. Magnusson. Multivariate Analysis of Orthogonal Range Searching and Graph Distances. Algorithmica, pages 1-24, 2020. Google Scholar
  15. H. Broersma, T. Kloks, D. Kratsch, and H. Müller. Independent sets in asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics, 12(2):276-287, 1999. Google Scholar
  16. S. Cabello. Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. ACM Transactions on Algorithms (TALG), 15(2):1-38, 2018. Google Scholar
  17. S. Cabello and C. Knauer. Algorithms for graphs of bounded treewidth via orthogonal range searching. Computational Geometry, 42(9):815-824, 2009. Google Scholar
  18. V. Chepoi, F. Dragan, and Y. Vaxès. Center and diameter problems in plane triangulations and quadrangulations. In Symposium on Discrete Algorithms (SODA'02), pages 346-355, 2002. Google Scholar
  19. V. Chepoi and F.F. Dragan. A linear-time algorithm for finding a central vertex of a chordal graph. In European Symposium on Algorithms, pages 159-170. Springer, 1994. Google Scholar
  20. V. Chepoi, F.F. Dragan, B. Estellon, M. Habib, and Y. Vaxès. Diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs. In Monique Teillaud, editor, Proceedings of the 24th ACM Symposium on Computational Geometry (SoCG), pages 59-68. ACM, 2008. Google Scholar
  21. V. Chepoi, F.F. Dragan, M. Habib, Y. Vaxès, and H. Alrasheed. Fast approximation of eccentricities and distances in hyperbolic graphs. Journal of Graph Algorithms and Applications, 23(2):393-433, 2019. Google Scholar
  22. V. Chepoi, B. Estellon, and Y. Vaxès. Covering planar graphs with a fixed number of balls. Discrete & Computational Geometry, 37(2):237-244, 2007. Google Scholar
  23. D. Corneil, F. Dragan, M. Habib, and C. Paul. Diameter determination on restricted graph families. Discrete Applied Mathematics, 113(2-3):143-166, 2001. Google Scholar
  24. D. Corneil, S. Olariu, and L. Stewart. Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM Journal on Computing, 28(4):1284-1297, 1999. Google Scholar
  25. D.G. Corneil, F.F. Dragan, and E. Köhler. On the power of BFS to determine a graph’s diameter. Networks: An International Journal, 42(4):209-222, 2003. Google Scholar
  26. D.G. Corneil, S. Olariu, and L. Stewart. A linear time algorithm to compute a dominating path in an AT-free graph. Information Processing Letters, 54(5):253-257, 1995. Google Scholar
  27. D.G. Corneil, S. Olariu, and L. Stewart. Asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics, 10(3):399-430, 1997. Google Scholar
  28. D.G. Corneil and J. Stacho. Vertex ordering characterizations of graphs of bounded asteroidal number. Journal of Graph Theory, 78(1):61-79, 2015. Google Scholar
  29. D. Coudert, G. Ducoffe, and A. Popa. Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. ACM Transactions on Algorithms (TALG), 15(3):1-57, 2019. Google Scholar
  30. M. Dalirrooyfard and J. Kaufmann. Approximation Algorithms for Min-Distance Problems in DAGs. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 60:1-60:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  31. C. Dallard, R. Ganian, M. Hatzel, M. Krnc, and M. Milanič. Graphs with two moplexes. In Proceedings of LAGOS'21, volume 195 of Procedia Computer Science, pages 248-256. Elsevier, 2021. Google Scholar
  32. J. De Rumeur. Communications dans les réseaux de processeurs. Masson Paris, 1994. Google Scholar
  33. F.F. Dragan. HT-graphs: centers, connected r-domination and Steiner trees. The Computer Science Journal of Moldova, 1(2):64-83, 1993. Google Scholar
  34. F.F. Dragan, G. Ducoffe, and H. Guarnera. Fast Deterministic Algorithms for Computing All Eccentricities in (Hyperbolic) Helly Graphs. In Anna Lubiw and Mohammad R. Salavatipour, editors, Algorithms and Data Structures (WADS), volume 12808 of Lecture Notes in Computer Science, pages 300-314. Springer, 2021. Google Scholar
  35. F.F. Dragan and E. Köhler. An approximation algorithm for the tree t-spanner problem on unweighted graphs via generalized chordal graphs. Algorithmica, 69(4):884-905, 2014. Google Scholar
  36. F.F. Dragan and F. Nicolai. LexBFS-orderings of distance-hereditary graphs with application to the diametral pair problem. Discrete Applied Mathematics, 98(3):191-207, 2000. Google Scholar
  37. F.F. Dragan, F. Nicolai, and A. Brandstädt. LexBFS-orderings and power of graphs. In Fabrizio d'Amore, Paolo Giulio Franciosa, and Alberto Marchetti-Spaccamela, editors, Graph-Theoretic Concepts in Computer Science (WG), volume 1197 of Lecture Notes in Computer Science, pages 166-180. Springer, 1996. Google Scholar
  38. G. Ducoffe. A New Application of Orthogonal Range Searching for Computing Giant Graph Diameters. In SOSA, 2019. Google Scholar
  39. G. Ducoffe. Beyond Helly Graphs: The Diameter Problem on Absolute Retracts. In Łukasz Kowalik, Michał Pilipczuk, and Paweł Rzażewski, editors, Graph-Theoretic Concepts in Computer Science (WG), pages 321-335. Springer International Publishing, 2021. Google Scholar
  40. G. Ducoffe. Isometric Embeddings in Trees and Their Use in Distance Problems. In Filippo Bonchi and Simon J. Puglisi, editors, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), volume 202 of Leibniz International Proceedings in Informatics (LIPIcs), pages 43:1-43:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  41. G. Ducoffe. Optimal centrality computations within bounded clique-width graphs. In 16th International Symposium on Parameterized and Exact Computation (IPEC), volume 214 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  42. G. Ducoffe. The diameter of AT-free graphs. Journal of Graph Theory, 99(4):594-614, 2022. Google Scholar
  43. G. Ducoffe and F.F. Dragan. A story of diameter, radius, and (almost) Helly property. Networks, 77(3):435-453, 2021. Google Scholar
  44. G. Ducoffe, M. Habib, and L. Viennot. Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1905-1922. SIAM, 2020. Google Scholar
  45. A. Farley and A. Proskurowski. Computation of the center and diameter of outerplanar graphs. Discrete Applied Mathematics, 2(3):185-191, 1980. Google Scholar
  46. F.V. Fomin, M. Matamala, E. Prisner, and I. Rapaport. AT-free graphs: linear bounds for the oriented diameter. Discrete Applied Mathematics, 141(1-3):135-148, 2004. Google Scholar
  47. T. Gallai. Transitiv orientierbare graphen. Acta Math. Hungarica, 18(1):25-66, 1967. Google Scholar
  48. F. Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974. Google Scholar
  49. P. Gawrychowski, H. Kaplan, S. Mozes, M. Sharir, and O. Weimann. Voronoi diagrams on planar graphs, and computing the diameter in deterministic Õ(n^5/3) time. In Symposium on Discrete Algorithms (SODA), pages 495-514. SIAM, 2018. Google Scholar
  50. P. Golovach, P. Heggernes, D. Kratsch, D. Lokshtanov, D. Meister, and S. Saurabh. Bandwidth on AT-free graphs. In International Symposium on Algorithms and Computation (ISAAC), pages 573-582. Springer, 2009. Google Scholar
  51. P.A. Golovach, D. Paulusma, and E.J. van Leeuwen. Induced disjoint paths in AT-free graphs. In Scandinavian Workshop on Algorithm Theory (SWAT), pages 153-164. Springer, 2012. Google Scholar
  52. M. C. Golumbic. Algorithmic graph theory and perfect graphs, volume 57. Elsevier, 2004. Google Scholar
  53. J. Gorzny and J. Huang. End-vertices of LBFS of (AT-free) bigraphs. Discrete Applied Mathematics, 225:87-94, 2017. Google Scholar
  54. M. Gromov. Hyperbolic Groups, pages 75-263. Springer, New York, NY, 1987. Google Scholar
  55. M. Habib, R. McConnell, C. Paul, and L. Viennot. Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science, 234(1-2):59-84, 2000. Google Scholar
  56. H. Hempel and D. Kratsch. On claw-free asteroidal triple-free graphs. Discrete Applied Mathematics, 121(1-3):155-180, 2002. Google Scholar
  57. R. Impagliazzo and R. Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  58. T. Kloks, D. Kratsch, and H. Müller. Asteroidal sets in graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 229-241. Springer, 1997. Google Scholar
  59. T. Kloks, D. Kratsch, and H. Müller. On the structure of graphs with bounded asteroidal number. Graphs and Combinatorics, 17(2):295-306, 2001. Google Scholar
  60. Dirk Koschützki, Katharina Anna Lehmann, Leon Peeters, Stefan Richter, Dagmar Tenfelde-Podehl, and Oliver Zlotowski. Centrality indices. In Network Analysis, pages 16-61. Springer, 2005. Google Scholar
  61. D. Kratsch and H. Müller. Colouring AT-free graphs. In European Symposium on Algorithms (ESA), pages 707-718. Springer, 2012. Google Scholar
  62. D. Kratsch, H. Müller, and I. Todinca. Feedback vertex set on AT-free graphs. Discrete Applied Mathematics, 156(10):1936-1947, 2008. Google Scholar
  63. D. Kratsch and J. Spinrad. Between O(nm) and O(n^α). SIAM Journal on Computing, 36(2):310-325, 2006. Google Scholar
  64. R. Laskar and D. Shier. On powers and centers of chordal graphs. Discrete Applied Mathematics, 6(2):139-147, 1983. Google Scholar
  65. I.-J. Lin, T.A. McKee, and D.B. West. The leafage of a chordal graph. Discussiones Mathematicae Graph Theory, 18(1):23-48, 1998. Google Scholar
  66. S. Olariu. A simple linear-time algorithm for computing the center of an interval graph. International Journal of Computer Mathematics, 34(3-4):121-128, 1990. Google Scholar
  67. R. Paige and R. Tarjan. Three partition refinement algorithms. SIAM Journal on Computing, 16(6):973-989, 1987. Google Scholar
  68. N. Pržulj, D.G. Corneil, and E. Köhler. Hereditary dominating pair graphs. Discrete Applied Mathematics, 134(1-3):239-261, 2004. Google Scholar
  69. L. Roditty and V. Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing (STOC), pages 515-524, 2013. Google Scholar
  70. D. Rose, R. Tarjan, and G. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5(2):266-283, 1976. Google Scholar
  71. R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci., 51(3):400-403, 1995. Google Scholar
  72. J. Stacho. 3-colouring AT-free graphs in polynomial time. In International Symposium on Algorithms and Computation (ISAAC), pages 144-155. Springer, 2010. Google Scholar
  73. L. Stewart and R. Valenzano. On polygon numbers of circle graphs and distance hereditary graphs. Discrete Applied Mathematics, 248:3-17, 2018. Google Scholar
  74. M. Tedder, D. Corneil, M. Habib, and C. Paul. Simpler linear-time modular decomposition via recursive factorizing permutations. In ICALP, pages 634-645. Springer, 2008. Google Scholar
  75. V. Vassilevska Williams and R.R. Williams. Subcubic equivalences between path, matrix, and triangle problems. Journal of the ACM (JACM), 65(5):1-38, 2018. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact