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The Longest Path Problem is Polynomial on Cocomparability Graphs

  • Kyriaki Ioannidou
  • Stavros D. Nikolopoulos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

The longest path problem is the problem of finding a path of maximum length in a graph. As a generalization of the Hamiltonian path problem, it is NP-complete on general graphs and, in fact, on every class of graphs that the Hamiltonian path problem is NP-complete. Polynomial solutions for the longest path problem have recently been proposed for weighted trees, ptolemaic graphs, bipartite permutation graphs, interval graphs, and some small classes of graphs. Although the Hamiltonian path problem on cocomparability graphs was proved to be polynomial almost two decades ago[9], the complexity status of the longest path problem on cocomparability graphs has remained open until now; actually, the complexity status of the problem has remained open even on the smaller class of permutation graphs. In this paper, we present a polynomial-time algorithm for solving the longest path problem on the class of cocomparability graphs. Our result resolves the open question for the complexity of the problem on such graphs, and since cocomparability graphs form a superclass of both interval and permutation graphs, extends the polynomial solution of the longest path problem on interval graphs[18] and provides polynomial solution to the class of permutation graphs.

Keywords

Longest path problem cocomparability graphs permutation graphs polynomial algorithm complexity 

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References

  1. 1.
    Arikati, S.R., Pandu Rangan, C.: Linear algorithm for optimal path cover problem on interval graphs. Inform. Proc. Lett. 35, 149–153 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asdre, K., Nikolopoulos, S.D.: The 1-fixed-endpoint path cover problem is polynomial on interval graphs. Algorithmica, doi:10.1007/s00453-009-9292-5Google Scholar
  3. 3.
    Bertossi, A.A.: Finding Hamiltonian circuits in proper interval graphs. Inform. Proc. Lett. 17, 97–101 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bulterman, R., van der Sommen, F., Zwaan, G., Verhoeff, T., van Gasteren, A., Feijen, W.: On computing a longest path in a tree. Inform. Proc. Lett. 81, 93–96 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chang, M.S., Peng, S.L., Liaw, J.L.: Deferred-query: An efficient approach for some problems on interval graphs. Networks 34, 1–10 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Damaschke, P.: The Hamiltonian circuit problem for circle graphs is NP-complete. Inform. Proc. Lett. 32, 1–2 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Damaschke, P.: Paths in interval graphs and circular arc graphs. Discrete Math. 112, 49–64 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Damaschke, P., Deogun, J.S., Kratsch, D., Steiner, G.: Finding Hamiltonian paths in cocomparability graphs using the bump number algorithm. Order 8, 383–391 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deogun, J.S., Steiner, G.: Polynomial algorithms for hamiltonian cycle in cocomparability graphs. SIAM J. Computing 23, 520–552 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Feder, T., Motwani, R.: Finding large cycles in Hamiltonian graphs. In: Proc. of the 16th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 166–175. ACM, New York (2005)Google Scholar
  12. 12.
    Gabow, H.N.: Finding paths and cycles of superpolylogarithmic length. In: Proc. of the 36th Annual ACM Symp. on Theory of Computing (STOC), pp. 407–416. ACM, New York (2004)Google Scholar
  13. 13.
    Gabow, H.N., Nie, S.: Finding long paths, cycles and circuits. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 752–763. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Computing 5, 704–714 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland Publishing Co., Amsterdam (2004)zbMATHGoogle Scholar
  17. 17.
    Habib, M., Mörhing, R.H., Steiner, G.: Computing the bump number is easy. Order 5, 107–129 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ioannidou, K., Mertzios, G.B., Nikolopoulos, S.D.: The longest path problem has a polynomial solution on interval graphs. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 403–414. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamiltonian paths in grid graphs. SIAM J. Computing 11, 676–686 (1982)CrossRefzbMATHGoogle Scholar
  20. 20.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  21. 21.
    Müller, H.: Hamiltonian circuits in chordal bipartite graphs. Discrete Math. 156, 291–298 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Narasimhan, G.: A note on the Hamiltonian circuit problem on directed path graphs. Inform. Proc. Lett. 32, 167–170 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Takahara, Y., Teramoto, S., Uehara, R.: Longest path problems on ptolemaic graphs. IEICE Trans. Inf. and Syst. 91-D, 170–177 (2008)CrossRefGoogle Scholar
  24. 24.
    Uehara, R.: Simple geometrical intersection graphs. In: Nakano, S.-i., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 25–33. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  25. 25.
    Uehara, R., Uno, Y.: Efficient algorithms for the longest path problem. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 871–883. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Uehara, R., Valiente, G.: Linear structure of bipartite permutation graphs and the longest path problem. Inform. Proc. Lett. 103, 71–77 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, Z., Li, H.: Algorithms for long paths in graphs. Theoret. Comput. Sci. 377, 25–34 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Kyriaki Ioannidou
    • 1
  • Stavros D. Nikolopoulos
    • 1
  1. 1.Department of Computer ScienceUniversity of IoanninaIoanninaGreece

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