Skip to main content
SpringerLink
Log in

A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

There has been intensive work on the parameterized complexity of the typically NP-hard task to edit undirected graphs into graphs fulfilling certain given vertex degree constraints. In this work, we lift the investigations to the case of directed graphs; herein, we focus on arc insertions. To this end, we develop a general two-stage framework which consists of efficiently solving a problem-specific number problem and transferring its solution to a solution for the graph problem by applying flow computations. In this way, we obtain fixed-parameter tractability and polynomial kernelizability results, with the central parameter being the maximum vertex in- or outdegree of the output digraph. Although there are certain similarities with the much better studied undirected case, the flow computation used in the directed case seems not to work for the undirected case while f-factor computations as used in the undirected case seem not to work for the directed case.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. All specific properties in this work can be easily decided in polynomial time. Indeed, in many cases even fixed-parameter tractability with respect to the maximum integer in the sequence would suffice.

  2. This special case was investigated more specifically in the Bachelor thesis of Koseler [30] (available online).

  3. Although not stated explicitly, the NP-hardness follows from the proof of Theorem 3.2 of the Bachelor thesis of Millani [35] (available online) as the construction therein allows for only one feasible target degree sequence.

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Upper Saddle River (1993)

    MATH  Google Scholar 

  2. Arora, S., Barak, B.: Complexity Theory: A Modern Approach. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  3. Bang-Jensen, J., Frank, A., Jackson, B.: Preserving and increasing local edge-connectivity in mixed graphs. SIAM J. Discret. Math. 8(2), 155–178 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bang-Jensen, J., Huang, J., Zhu, X.: Completing orientations of partially oriented graphs. J. Graph Theory 87(3), 285–304 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bazgan, C., Bredereck, R., Hartung, S., Nichterlein, A., Woeginger, G.J.: Finding large degree-anonymous subgraphs is hard. Theor. Comput. Sci. 622, 90–110 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas-Roma, J., Herrera-Joancomartí, J., Torra, V.: \(k\)-degree anonymity and edge selection: improving data utility in large networks. Knowl. Inf. Syst. 50(2), 447–474 (2017)

    Article  Google Scholar 

  7. Chen, W.-K.: On the realization of a \((p, s)\)-digraph with prescribed degrees. J. Frankl. Inst. 281(5), 406–422 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chester, S., Kapron, B., Srivastava, G., Venkatesh, S.: Complexity of social network anonymization. Soc. Netw. Anal. Min. 3(2), 151–166 (2013)

    Article  Google Scholar 

  9. Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M., Schlotter, I.: Parameterized complexity of eulerian deletion problems. Algorithmica 68(1), 41–61 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  11. Dorn, F., Moser, H., Niedermeier, R., Weller, M.: Efficient algorithms for eulerian extension and rural postman. SIAM J. Discret. Math. 27(1), 75–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  13. Erdős, P., Gallai, T.: Graphs with prescribed degrees of vertices (in Hungarian). Mat. Lapok 11, 264–274 (1960)

    MATH  Google Scholar 

  14. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  15. Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Froese, V., Nichterlein, A., Niedermeier, R.: Win-win kernelization for degree sequence completion problems. J. Comput. Syst. Sci. 82(6), 1100–1111 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fulkerson, D.: Zero-one matrices with zero trace. Pac. J. Math. 10(3), 831–836 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gale, D.: A theorem on flows in networks. Pac. J. Math. 7, 1073–1082 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  19. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  20. Golovach, P.A.: Editing to a graph of given degrees. Theor. Comput. Sci. 591, 72–84 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Golovach, P.A., Mertzios, G.B.: Graph editing to a given degree sequence. Theor. Comput. Sci. 665, 1–12 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gutin, G., Yeo, A.: Some parameterized problems on digraphs. Comput. J. 51(3), 363–371 (2008)

    Article  Google Scholar 

  23. Hakimi, S.: On realizability of a set of integers as degrees of the vertices of a linear graph. I. J. SIAM 10(3), 496–506 (1962)

    MathSciNet  MATH  Google Scholar 

  24. Hartung, S., Nichterlein, A.: NP-hardness and fixed-parameter tractability of realizing degree sequences with directed acyclic graphs. SIAM J. Discret. Math. 29(4), 1931–1960 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hartung, S., Hoffmann, C., Nichterlein, A.: Improved upper and lower bound heuristics for degree anonymization in social networks. In: Proceedings of the 13th international symposium on experimental algorithms (SEA ’14), vol. 8504 of LNCS, pp. 376–387. Springer (2014)

  26. Hartung, S., Nichterlein, A., Niedermeier, R., Suchý, O.: A refined complexity analysis of degree anonymization in graphs. Inf. Comput. 243, 249–262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kleitman, D., Wang, D.: Algorithms for constructing graphs and digraphs with given valences and factors. SIAM J. Discret. Math. 6(1), 79–88 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  30. Koseler, M.: Kernelization for degree-constraint editing on directed graphs. Bachelor thesis, TU Berlin (2015). URL http://fpt.akt.tu-berlin.de/publications/theses/BA-marcel-koseler.pdf

  31. Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  32. Leskovec, J., Horvitz, E.: Planetary-scale views on a large instant-messaging network. In: Proceedings of the 17th international conference on World Wide Web (WWW ’08), ACM, pp. 915–924 (2008)

  33. Liu, K., Terzi, E.: Towards identity anonymization on graphs. In: Proceedings of the ACM SIGMOD international conference on management of data, SIGMOD ’08, ACM, pp. 93–106 (2008)

  34. Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: a parameterized approach. J. Comput. Syst. Sci. 78(1), 179–191 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Millani, M.G.: Algorithms and complexity for degree anonymization in directed graphs. Bachelor thesis, TU Berlin (2015). URL http://fpt.akt.tu-berlin.de/publications/theses/BA-marcelo-millani.pdf

  36. Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. J. Discret. Algorithms 7(2), 181–190 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  38. Orlin, J.B.: Max flows in O(nm) time, or better. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC ’13), ACM, pp. 765–774 (2013)

  39. Weller, M., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: On making directed graphs transitive. J. Comput. Syst. Sci. 78(2), 559–574 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    Robert Bredereck, Vincent Froese, Marcel Koseler, Marcelo Garlet Millani, André Nichterlein & Rolf Niedermeier

Authors
  1. Robert Bredereck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vincent Froese
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcel Koseler
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marcelo Garlet Millani
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. André Nichterlein
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Rolf Niedermeier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Froese.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A short version of this article appeared in the Proceedings of the 11th International Symposium on Parameterized and Exact Computation (IPEC ’16), pp. 10:1–10:14. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bredereck, R., Froese, V., Koseler, M. et al. A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs. Algorithmica 81, 1584–1614 (2019). https://doi.org/10.1007/s00453-018-0494-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-018-0494-6

Keywords

Search

Navigation