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The Longest Path Problem has a Polynomial Solution on Interval Graphs

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Abstract

The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno (Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC), LNCS, vol. 3341, pp. 871–883, 2004), where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm uses a dynamic programming approach and runs in O(n 4) time, where n is the number of vertices of the input graph.

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References

  1. Arikati, S.R., Pandu Rangan, C.: Linear algorithm for optimal path cover problem on interval graphs. Inf. Process. Lett. 35, 149–153 (1990)

    Article  MATH  Google Scholar 

  2. Asdre, K., Nikolopoulos, S.D.: The 1-fixed-endpoint path cover problem is polynomial on interval graphs. Algorithmica (2009). doi:10.1007/s00453-009-9292-5

    Google Scholar 

  3. Bertossi, A.A.: Finding Hamiltonian circuits in proper interval graphs. Inf. Process. Lett. 17, 97–101 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bulterman, R., van der Sommen, F., Zwaan, G., Verhoeff, T., van Gasteren, A., Feijen, W.: On computing a longest path in a tree. Inf. Process. Lett. 81, 93–96 (2002)

    Article  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Damaschke, P.: The Hamiltonian circuit problem for circle graphs is NP-complete. Inf. Process. Lett. 32, 1–2 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Damaschke, P.: Paths in interval graphs and circular arc graphs. Discrete Math. 112, 49–64 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. 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 

  11. Gabow, H.N., Nie, S.: Finding long paths, cycles and circuits. In: Proc. of the 19th Annual International Symp. on Algorithms and Computation (ISAAC). LNCS, vol. 5369, pp. 752–763. Springer, Berlin (2008)

    Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  13. Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5, 704–714 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. J. Comput. Biol. 2, 139–152 (1995)

    Article  Google Scholar 

  15. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland, Amsterdam (2004)

    MATH  Google Scholar 

  16. Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamiltonian paths in grid graphs. SIAM J. Comput. 11, 676–686 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Karger, D., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18, 82–98 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Keil, J.M.: Finding Hamiltonian circuits in interval graphs. Inf. Process. Lett. 20, 201–206 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Müller, H.: Hamiltonian circuits in chordal bipartite graphs. Discrete Math. 156, 291–298 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Narasimhan, G.: A note on the Hamiltonian circuit problem on directed path graphs. Inf. Process. Lett. 32, 167–170 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ramalingam, G., Rangan, C. Pandu: A unified approach to domination problems on interval graphs. Inf. Process. Lett. 27, 271–274 (1988)

    Article  MATH  Google Scholar 

  22. Takahara, Y., Teramoto, S., Uehara, R.: Longest path problems on Ptolemaic graphs. IEICE Trans. Inf. Syst. 91-D, 170–177 (2008)

    Article  Google Scholar 

  23. Uehara, R., Uno, Y.: Efficient algorithms for the longest path problem. In: Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC). LNCS, vol. 3341, pp. 871–883. Springer, Berlin (2004)

    Google Scholar 

  24. Uehara, R., Valiente, G.: Linear structure of bipartite permutation graphs and the longest path problem. Inf. Process. Lett. 103, 71–77 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vishwanathan, S.: An approximation algorithm for finding a long path in Hamiltonian graphs. In: Proc. of the 11th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 680–685. ACM, New York (2000)

    Google Scholar 

  26. Zhang, Z., Li, H.: Algorithms for long paths in graphs. Theoret. Comput. Sci. 377, 25–34 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Ioannina, P.O. Box 1186, 45110, Ioannina, Greece

    Kyriaki Ioannidou & Stavros D. Nikolopoulos

  2. Department of Computer Science, RWTH Aachen University, Ahornstr. 55, 52074, Aachen, Germany

    George B. Mertzios

Authors
  1. Kyriaki Ioannidou
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  2. George B. Mertzios
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  3. Stavros D. Nikolopoulos
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Corresponding author

Correspondence to Stavros D. Nikolopoulos.

Additional information

K. Ioannidou was co-financed by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%).

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Ioannidou, K., Mertzios, G.B. & Nikolopoulos, S.D. The Longest Path Problem has a Polynomial Solution on Interval Graphs. Algorithmica 61, 320–341 (2011). https://doi.org/10.1007/s00453-010-9411-3

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  • DOI: https://doi.org/10.1007/s00453-010-9411-3

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