Computing the Largest Bond of a Graph

Authors Gabriel L. Duarte, Daniel Lokshtanov, Lehilton L. C. Pedrosa , Rafael C. S. Schouery , Uéverton S. Souza



PDF
Thumbnail PDF

File

LIPIcs.IPEC.2019.12.pdf
  • Filesize: 0.5 MB
  • 15 pages

Document Identifiers

Author Details

Gabriel L. Duarte
  • Fluminense Federal University, Rio de Janeiro, Brazil
Daniel Lokshtanov
  • University of California Santa Barbara, CA, USA
Lehilton L. C. Pedrosa
  • University of Campinas, São Paulo, Brazil
Rafael C. S. Schouery
  • University of Campinas, São Paulo, Brazil
Uéverton S. Souza
  • Fluminense Federal University, Rio de Janeiro, Brazil

Acknowledgements

We thank the organizers of WoPOCA 2017 for the opportunity to bring together some of the co-authors of this paper.

Cite AsGet BibTex

Gabriel L. Duarte, Daniel Lokshtanov, Lehilton L. C. Pedrosa, Rafael C. S. Schouery, and Uéverton S. Souza. Computing the Largest Bond of a Graph. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2019.12

Abstract

A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts [S,V\S] of G such that G[S] and G[V\S] are both connected. Given s,t in V(G), an st-bond of G is a bond whose removal disconnects s and t. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest st-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that Largest Bond remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless P = NP. We also show that Largest Bond and Largest st-Bond on graphs of clique-width w cannot be solved in time f(w) x n^{o(w)} unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) x n^{O(w)}. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP subseteq coNP/poly.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • bond
  • cut
  • maximum cut
  • connected cut
  • FPT
  • treewidth
  • clique-width

Metrics

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

References

  1. Robert E. L. Aldred, Dries V. Dyck, Gunnar Brinkmann, Veerle Fack, and Brendan D McKay. Graph structural properties of non-Yutsis graphs allowing fast recognition. Discrete Applied Mathematics, 157(2):377-386, 2009. Google Scholar
  2. Lawrence Christian Biedenharn and James D Louck. The Racah-Wigner algebra in quantum theory. Addison-Wesley, 1981. Google Scholar
  3. Hans L Bodlaender, Jan Van Leeuwen, Richard Tan, and Dimitrios M Thilikos. On interval routing schemes and treewidth. Information and Computation, 139(1):92-109, 1997. Google Scholar
  4. Jung Jin Cho, Yong Chen, and Yu Ding. On the (co)girth of a connected matroid. Discrete Applied Mathematics, 155(18):2456-2470, 2007. Google Scholar
  5. T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein. Introduction To Algorithms. MIT Press, 2001. URL: https://books.google.co.in/books?id=NLngYyWFl_YC.
  6. Bruno Courcelle, Johann A Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. Google Scholar
  7. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. Google Scholar
  8. Marek Cygan, Fedor V Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 4. Springer, 2015. Google Scholar
  9. Guoli Ding, Stan Dziobiak, and Haidong Wu. Large-or-Minors in 3-Connected Graphs. Journal of Graph Theory, 82(2):207-217, 2016. Google Scholar
  10. Dries V. Dyck and Veerle Fack. On the reduction of Yutsis graphs. Discrete Mathematics, 307(11):1506-1515, 2007. The Fourth Caracow Conference on Graph Theory. Google Scholar
  11. Jack Edmonds and Richard M. Karp. Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems. J. ACM, 19(2):248-264, April 1972. Google Scholar
  12. Paul Erdös. On some extremal problems in graph theory. Israel Journal of Mathematics, 3(2):113-116, 1965. Google Scholar
  13. Melissa Flynn. The Largest Bond in 3-Connected Graphs. PhD thesis, The University of Mississippi, 2017. Google Scholar
  14. Fedor V Fomin, Petr A Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM Journal on Computing, 43(5):1541-1563, 2014. Google Scholar
  15. Rajiv Gandhi, Mohammad T. Hajiaghayi, Guy Kortsarz, Manish Purohit, and Kanthi Sarpatwar. On maximum leaf trees and connections to connected maximum cut problems. Information Processing Letters, 129:31-34, 2018. Google Scholar
  16. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  17. Michael R. Garey, David S. Johnson, and Robert E. Tarjan. The Planar Hamiltonian Circuit Problem is NP-Complete. SIAM Journal on Computing, 5(4):704-714, 1976. Google Scholar
  18. Christopher D. Godsil and Gordon F. Royle. Algebraic Graph Theory. Graduate texts in mathematics. Springer, 2001. Google Scholar
  19. Michel X. Goemans and David P. Williamson. Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. J. ACM, 42(6):1115-1145, 1995. Google Scholar
  20. Ralph E Gomory and Tien Chung Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. Google Scholar
  21. F. Hadlock. Finding a Maximum Cut of a Planar Graph in Polynomial Time. SIAM Journal on Computing, 4(3):221-225, 1975. Google Scholar
  22. D. J. Haglin and S. M. Venkatesan. Approximation and intractability results for the maximum cut problem and its variants. IEEE Transactions on Computers, 40(1):110-113, January 1991. URL: https://doi.org/10.1109/12.67327.
  23. Mohammad Taghi Hajiaghayi, Guy Kortsarz, Robert MacDavid, Manish Purohit, and Kanthi Sarpatwar. Approximation Algorithms for Connected Maximum Cut and Related Problems. In In Proceedings of the 23rd European Symposium on Algorithms, pages 693-704, 2015. Google Scholar
  24. S. Khot, G. Kindler, E. Mossel, and R. O’Donnell. Optimal Inapproximability Results for MAX‐CUT and Other 2‐Variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007. Google Scholar
  25. Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms, 31(2):335-354, 1999. Google Scholar
  26. Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005. Google Scholar
  27. Andrew M Odlyzko. Asymptotic enumeration methods. Handbook of combinatorics, 2(1063):1229, 1995. Google Scholar
  28. James G. Oxley. Matroid theory, volume 3. Oxford University Press, USA, 2006. Google Scholar
  29. Christophe Picouleau. Complexity of the hamiltonian cycle in regular graph problem. Theoretical Computer Science, 131(2):463-473, 1994. Google Scholar
  30. Saket Saurabh and Meirav Zehavi. Parameterized Complexity of Multi-Node Hubs. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018), pages 8:1-8:14, 2019. Google Scholar
  31. Xiangkun Shen, Jon Lee, and Viswanath Nagarajan. Approximating graph-constrained max-cut. Mathematical Programming, 172(1):35-58, November 2018. Google Scholar
  32. A. P. Yutsis, V. V. Vanagas, and I. B. Levinson. Mathematical apparatus of the theory of angular momentum. Israel program for scientific translations, 1962. 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