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Complexity of Linear Connectivity Problems in Directed Hypergraphs

  • Mayur Thakur
  • Rahul Tripathi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3328)

Abstract

We introduce a notion of linear hyperconnection (formally denoted L-hyperpath) between nodes in a directed hypergraph and relate this notion to existing notions of hyperpaths in directed hypergraphs. We observe that many interesting questions in problem domains such as secret transfer protocols, routing in packet filtered networks, and propositional satisfiability are basically questions about existence of L-hyperpaths or about cyclomatic number of directed hypergraphs w.r.t. L-hypercycles (the minimum number of hyperedges that need to be deleted to make a directed hypergraph free of L-hypercycles). We prove that the L-hyperpath existence problem, the cyclomatic number problem, the minimum cyclomatic set problem, and the minimal cyclomatic set problem are each complete for a different level (respectively, NP, \({\it \Sigma}^{p}_{2}\), \({\it \Pi}^{p}_{2}\), and DP) of the polynomial hierarchy.

Keywords

Directed Graph Problem Domain Simple Path Simple Cycle Reachability Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Acharya, B.: On the cyclomatic number of a hypergraph. Discrete Mathematics 27, 111–116 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alimonti, P., Feuerstein, E., Nanni, U.: Linear time algorithms for liveness and boundedness in conflict-free Petri nets. In: Simon, I. (ed.) LATIN 1992. LNCS, vol. 583, pp. 1–14. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  3. 3.
    Ausiello, G., D’Atri, A., Saccá, D.: Graph algorithms for functional dependency manipulation. Journal of the ACM 30, 752–766 (1983)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ausiello, G., D’Atri, A., Saccá, D.: Minimal representation of directed hypergraphs. SIAM Journal on Computing 15, 418–431 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ausiello, G., Franciosa, P., Frigioni, D.: Directed hypergraphs: Problems, algorithmic results, and a novel decremental approach. In: Restivo, A., Ronchi Della Rocca, S., Roversi, L. (eds.) ICTCS 2001. LNCS, vol. 2202, pp. 312–328. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Ausiello, G., Giaccio, R.: On-line algorithms for satisfiability formulae with uncertainty. Theoretical Computer Science 171, 3–24 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ausiello, G., Giaccio, R., Italiano, G., Nanni, U.: Optimal traversal of directed hypergraphs (Manuscript 1997)Google Scholar
  8. 8.
    Ausiello, G., Italiano, G., Nanni, U.: Hypergraph traversal revisited: Cost measures and dynamic algorithms. In: Brim, L., Gruska, J., Zlatuška, J. (eds.) MFCS 1998. LNCS, vol. 1450, pp. 1–16. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  10. 10.
    Gallo, G., Longo, G., Pallottino, S., Nguyen, S.: Directed hypergraphs and applications. Discrete Applied Mathematics 42, 177–201 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gallo, G., Rago, G.: A hypergraph approach to logical inference for datalog formulae. Technical Report 28/90, Dip. di Informatica, Univ. of Pisa, Italy (1990)Google Scholar
  12. 12.
    Gallo, G., Scutella, M.: Directed hypergraphs as a modelling paradigm. Technical Report TR-99-02, Dipartimento di Informatica (February 1999)Google Scholar
  13. 13.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Information and Control 56(3), 183–198 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Italiano, G., Nanni, U.: On line maintainenance of minimal directed hypergraphs. In: 3rd Italian Conf. on Theoretical Computer Science, pp. 335–349. World Scientific Co., Singapore (1989)Google Scholar
  16. 16.
    Klein, D., Manning, C.: Parsing and hypergraphs. In: Proceedings of the 7th International Workshop on Parsing Technologies, IWPT 2001(2001) Google Scholar
  17. 17.
    Knuth, D.: A generalization of Dijkstra’s algorithm. Information Processing Letters 6(1), 1–5 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Nguyen, S., Pallottino, S.: Hyperpaths and shortest hyperpaths. Combinatorial Optimization 1403, 258–271 (1989)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Nielsen, L., Pretolani, D., Andersen, K.: A remark on the definition of a B-hyperpath. Technical report, Department of Operations Research, University of Aarhus (2001)Google Scholar
  20. 20.
    Nilson, N.: Principles of Artificial Intelligence. Springer, Heidelberg (1982)Google Scholar
  21. 21.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  22. 22.
    Papadimitriou, C., Yannakakis, M.: A note on succinct representations of graphs. Information and Control 71(3), 181–185 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Petri, C.: Communication with automata. Technical Report Supplement 1 to Tech. Report RADC-TR-65-377,1, Univ. of Bonn (1962)Google Scholar
  24. 24.
    Ramalingam, G., Reps, T.: An incremental algorithm for a generalization of the Shortest Path problem. Journal of Algorithms 21, 267–305 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tantau, T.: A note on the complexity of the reachability problem for tournaments. In: ECCCTR: Electronic Colloquium on Computational Complexity (2001)Google Scholar
  26. 26.
    Temkin, O., Zeigarnik, A., Bonchev, D.: Chemical Reaction Networks: A Graph-Theoretical Approach. CRC Press, Boca Raton (1996)Google Scholar
  27. 27.
    Thakur, M., Tripathi, R.: Complexity of linear connectivity problems in directed hypergraphs. Technical Report TR814, Department of Computer Science, University of Rochester (September 2003)Google Scholar
  28. 28.
    Ullman, J.: Principles of Database Systems. Computer Science Press (1982)Google Scholar
  29. 29.
    Wagner, K.: The complexity of combinatorial problems with succinct input representations. Acta Informatica 23, 325–356 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zeigarnik, A.: On hypercycles and hypercircuits in hypergraphs. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 51 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Mayur Thakur
    • 1
  • Rahul Tripathi
    • 2
  1. 1.Dept. of Computer ScienceUniversity of Missouri–RollaRollaUSA
  2. 2.Dept. of Computer ScienceUniversity of RochesterRochesterUSA

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