Skip to main content
SpringerLink
Log in

Complexity Classification of Some Edge Modification Problems

  • Conference paper
Graph-Theoretic Concepts in Computer Science (WG 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1665))

Included in the following conference series:

Abstract

In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove in this paper the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Agrawal, P. Klein, and R. Ravi. Cutting down on fill using nested dissection: provably good elimination orderings. In A. George, J. R. Gilbert, and J. W. H. Liu, editors, Graph Theory and Sparse Matrix Computation, pages 31–55. Springer, 1993. 67

    Google Scholar 

  2. T. Asano. An application of duality to edge-deletion problems. SIAM Journal on Computing, 16(2):312–331, 1987. 66

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Asano and T. Hirata. Edge-deletion and edge-contraction problems. In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pages 245–254, San Francisco, California, 5-7 May 1982. 66

    Google Scholar 

  4. A. Ben-Dor. Private communication, 1996. 66

    Google Scholar 

  5. H. Bodlaender and B. de Fluiter. On intervalizing k-colored graphs for DNA physical mapping. Discrete Applied Math., 71:55–77, 1996. 65, 66

    Article  MATH  Google Scholar 

  6. A. Brandstädt, V. B. Le, and J. P. Spinrad. Graph Classes-a Survey. SIAM, Philadelphia, 1999. SIAM Monographs in Discrete Mathematics and Applications. 69

    Google Scholar 

  7. L. Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58:171–176, 1996. 66

    Article  MATH  MathSciNet  Google Scholar 

  8. K. Cirino, S. Muthukrishnan, N. Narayanaswamy, and H. Ramesh. Graph editing to bipartite interval graphs: exact and asymptotic bounds. Technical report, Bell Laboratories Innovations, Lucent Technologies, 1996. 65, 66, 67

    Google Scholar 

  9. D. G. Corneil, S. Olariu, and L. Stewart. The linear structure of graphs: asteroidal triple-free graphs. In Proc. 19th Int. Workshop (WG’ 93), Graph-Theoretic Concepts in Computer Science, pages 211–224. Springer-Verlag, 1994. LNCS 790. 73

    Google Scholar 

  10. El-Mallah and Colbourn. The complexity of some edge deletion problems. IEEE Transactions on Circuits and Systems, 35(3):354–362, 1988. 66, 66

    Article  MATH  MathSciNet  Google Scholar 

  11. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., San Francisco, 1979. 65, 66

    MATH  Google Scholar 

  12. P. W. Goldberg, M. C. Golumbic, H. Kaplan, and R. Shamir. Four strikes against physical mapping of DNA. Journal of Computational Biology, 2(1):139–152, 1995. 65, 66, 66

    Article  Google Scholar 

  13. M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980. 69

    MATH  Google Scholar 

  14. M. C. Golumbic, H. Kaplan, and R. Shamir. On the complexity of DNA physical mapping. Advances in Applied Mathematics, 15:251–261, 1994. 65

    Article  MATH  MathSciNet  Google Scholar 

  15. M. C. Golumbic, H. Kaplan, and R. Shamir. Graph sandwich problems. Journal of Algorithms, 19:449–473, 1995. 66

    Article  MATH  MathSciNet  Google Scholar 

  16. M. C. Golumbic and R. Shamir. Complexity and algorithms for reasoning about time: A graph-theoretic approach. J. ACM, 40:1108–1133, 1993. 66

    Article  MATH  MathSciNet  Google Scholar 

  17. S. L. Hakimi, E. F. Schmeichel, and N. E. Young. Orienting graphs to optimize reachability. Information Processing Letters, 63(5):229–235, 1997. 66

    Article  MathSciNet  Google Scholar 

  18. P. L. Hammer, T. Ibaraki, and U. N. Peled. Threshold numbers and threshold completions. In P. Hansen, editor, Studies on Graphs and Discrete Programming, pages 125–145. North-Holland, 1981. 66

    Google Scholar 

  19. P. L. Hammer and B. Simeone. The splittance of a graph. Combinatorica, 1:275–284, 1981. 66

    Article  MATH  MathSciNet  Google Scholar 

  20. J. Håstad. Some optimal inapproximability results. In Proc. 29th STOC, pages 1–10, 1997. Full version: E-CCC Report number TR97-037. 74

    Google Scholar 

  21. H. Kaplan and R. Shamir. Physical maps and interval sandwich problems: Bounded degrees help. In Proc. ISTCS, pages 195–201, 1996. 66, 75

    Google Scholar 

  22. H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780–791. IEEE Computer Science Press, Los Alamitos, California, 1994. to appear in SIAM J. Computing. 66

    Chapter  Google Scholar 

  23. T. Kashiwabara and T. Fujisawa. An NP-complete problem on interval graphs. In IEEE International Symposium on Circuits and Systems (12th), pages 82–83, 1979. 66

    Google Scholar 

  24. J. Lewis and M. Yannakakis. The node deletion problem for hereditary properties is NP-complete. J. Comput. Sys. Sci., 20:219–230, 1980. 66, 75

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Lovás. A characterization of perfect graphs. J. Combin. Theory, pages 95–98, 1972. 71

    Google Scholar 

  26. C. Lund and M. Yannakakis. The approximation of maximum subgraph problems. In A. Lingas, R. Karlsson, and S. Carlsson, editors, Proceedings of International Conference on Automata, Languages and Programming (ICALP’ 91), pages 40–51, Berlin, Germany, 1993. Springer. LNCS 700. 66, 74

    Google Scholar 

  27. F. Margot. Some complexity results about threshold graphs. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science, 49, 1994. 66

    Google Scholar 

  28. A. Natanzon, R. Shamir, and R. Sharan. A polynomial approximation algorithm for the minimum fill-in problem. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC’98), pages 41–47, New York, May 23-26 1998. ACM Press. 66

    Google Scholar 

  29. R. Ravi, A. Agrawal, and P. Klein. Ordering problems approximated: single processor scheduling and interval graph completion. In Proc. ICALP 1991, pages 751–762. Springer, 1991. LNCS 510. 67

    Google Scholar 

  30. J. D. Rose. A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In R. C. Reed, editor, Graph Theory and Computing, pages 183–217. Academic Press, N.Y., 1972. 66

    Google Scholar 

  31. R. E. Tarjan and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, text acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Computing, 13:566–579, 1984. 72

    Article  MATH  MathSciNet  Google Scholar 

  32. L. Trevisan, G. Sorkin, M. Sudan, and D. Williamson. Gadgets, approximation, and linear programming. In Proc. IEEE Symposium on Foundations of Computer Science (FOCS’96), pages 617–626, 1996. 74

    Google Scholar 

  33. J. Xue. Edge-maximal triangulated subgraph and heuristics for the maximum clique problem. Technical report, Graduate School of Management, Clark University, Worcester, MA, July 1993. 66

    Google Scholar 

  34. M. Yannakakis. Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth., 2, 1981. 66, 66, 70, 70

    Google Scholar 

  35. M. Yannakakis. Edge deletion problems. SIAM J. Computing, 10(2):297–309, 1981. 66

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Tel Aviv University, Tel-Aviv, Israel

    Assaf Natanzon, Ron Shamir & Roded Sharan

Authors
  1. Assaf Natanzon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ron Shamir
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roded Sharan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Theoretical Computer Science ETH Zürich ETH Zentrum, 8092, Zürich, Switzerland

    Peter Widmayer , Gabriele Neyer  & Stephan Eidenbenz ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Natanzon, A., Shamir, R., Sharan, R. (1999). Complexity Classification of Some Edge Modification Problems. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds) Graph-Theoretic Concepts in Computer Science. WG 1999. Lecture Notes in Computer Science, vol 1665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46784-X_8

Download citation

  • DOI: https://doi.org/10.1007/3-540-46784-X_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66731-5

  • Online ISBN: 978-3-540-46784-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation