Efficient Algorithms for the Longest Path Problem

  • Ryuhei Uehara
  • Yushi Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3341)


The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, very few graph classes are known where the longest path problem can be solved efficiently. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and it then solves the longest path problem efficiently for weighted trees, block graphs, ptolemaic graphs, and cacti. We next propose three new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on those classes. As a corollary, it is also shown that the problem can be solved efficiently on threshold graphs.


Efficient algorithms graph classes longest path problem 


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  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-Coding. J. ACM 42(4), 844–856 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  3. 3.
    Balister, P.N., Györi, E., Lehel, J., Schelp, R.H.: Longest Paths in Circular Arc Graphs. Technical report, U. of Memphis (2002),
  4. 4.
    Berge, C.: Hypergraphs. Elsevier, Amsterdam (1989)zbMATHGoogle Scholar
  5. 5.
    Bertossi, A.A.: Finding Hamiltonian Circuits in Proper Interval Graphs. Info. Proc. Lett. 17(2), 97–101 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bertossi, A.A., Bonuccelli, M.A.: Hamiltonian Circuits in Interval Graph Generalizations. Info. Proc. Lett. 23, 195–200 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Björklund, A., Husfeldt, T.: Finding a Path of Superlogarithmic Length. SIAM J. Comput. 32(6), 1395–1402 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Booth, K.S., Lueker, G.S.: Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)zbMATHCrossRefGoogle Scholar
  10. 10.
    Bulterman, R.W., van der Sommen, F.W., Zwaan, G., Verhoeff, T., van Gasteren, A.J.M., Feijen, W.H.J.: On Computing a Longest Path in a Tree. Info. Proc. Lett. 81, 93–96 (2002)zbMATHCrossRefGoogle Scholar
  11. 11.
    Damaschke, P.: The Hamiltonian Circuit Problem for Circle Graphs is NP-complete. Info. Proc. Lett. 32, 1–2 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Damaschke, P.: Paths in Interval Graphs and Circular Arc Graphs. Discr. Math. 112, 49–64 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  14. 14.
    Gabow, H.N.: Data Structures for Weighted Matching and Nearest Common Ancestors with Linking. In: Proc. 1st Ann. ACM-SIAM Symp. on Discr. Algo., pp. 434–443. ACM, New York (1990)Google Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability — A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  16. 16.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. In: Ann. Discr. Math., vol. 57, 2/e. Elsevier, Amsterdam (2004)Google Scholar
  17. 17.
    Hochbaum, D.: Approximation Algorithms for NP-hard Problems. PWS Publishing Company (1995)Google Scholar
  18. 18.
    Karger, D., Motwani, R., Ramkumar, G.D.S.: On Approximating the Longest Path in a Graph. Algorithmica 18, 82–98 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Korte, N., Möhring, R.H.: An Incremental Linear-Time Algorithm for Recognizing Interval Graphs. SIAM J. Comput 18(1), 68–81 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Mahadev, N.V.R., Peled, U.N.: Longest Cycles in Threshold Graphs. Discr. Math. 135, 169–176 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Monien, B.: How to Find Long Paths Efficiently. Ann. Discr. Math. 25, 239–254 (1985)MathSciNetGoogle Scholar
  22. 22.
    Müller, H.: Hamiltonian Circuit in Chordal Bipartite Graphs. Disc. Math. 156, 291–298 (1996)zbMATHCrossRefGoogle Scholar
  23. 23.
    Scutellà, M.G.: An Approximation Algorithm for Computing Longest Paths. European J. Oper. Res. 148(3), 584–590 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite Permutation Graphs. Discr. Appl. Math. 18, 279–292 (1987)zbMATHCrossRefGoogle Scholar
  25. 25.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  26. 26.
    Vishwanathan, S.: An Approximation Algorithm for Finding a Long Path in Hamiltonian Graphs. In: Proc. 11th Ann. ACM-SIAM Symp. on Discr. Algo., pp. 680–685. ACM, New York (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ryuhei Uehara
    • 1
  • Yushi Uno
    • 2
  1. 1.Department of Information Processing, School of Information ScienceJAISTIshikawaJapan
  2. 2.Department of Mathematics and Information Science, College of Integrated Arts and SciencesOsaka Prefecture UniversitySakaiJapan

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