Graph inference from a walk for trees of bounded degree 3 is NP-complete

  • Osamu MaruyamaEmail author
  • Satoru Miyano
Contributed Papers Graphs in Models of Computations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 969)


The graph inference from a walk for a class C of undirected edge-colored graphs is, given a string x of colors, finding the smallest graph G in C that allows a traverse of all edges in G whose sequence of edge-colors is x, called a walk for x. We prove that the graph inference from a walk for trees of bounded degree k is NP-complete for any k ⩾ 3, while the problem for trees without any degree bound constraint is known to be solvable in O(n) time, where n is the length of the string. Furthermore, the problem for a special class of trees of bounded degree 3, called (1,1)-caterpillars, is shown to be NP-complete. This contrast with the case that the problem for linear chains is known to be solvable in O(n log n) time since a (1,1)-caterpillar is obtained by attaching at most one hair of length one to each node of a linear chain. We also show the MAXSNP-hardness of these problems.


Polynomial Time Small Tree Linear Chain Vertex Cover Polynomial Time Approximation Scheme 
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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  1. 1.
    D. Angluin. On the complexity of minimum inference of regular sets. Inform. Control, 39:337–350, 1978.CrossRefGoogle Scholar
  2. 2.
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems. In Proc. 33rd IEEE Symp. Foundations of Computer Science, pages 14–23, 1992.Google Scholar
  3. 3.
    J. A. Aslam and R. L. Rivest. Inferring graphs from walks. In Proc. 3rd Workshop on Computational Learning Theory, pages 359–370, 1990.Google Scholar
  4. 4.
    M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979.Google Scholar
  5. 5.
    E. M. Gold. Complexity of automaton identification from given data. Inform. Control, 37:302–320, 1978.Google Scholar
  6. 6.
    J. Haralambides, F. Makedon, and B. Monien. Badndwidth minimization: An approximation algorithm for caterpillars. Math. Systems Theory, 24:169–177, 1991.Google Scholar
  7. 7.
    O. Maruyama and S. Miyano. Inferring a tree from walks. In Proc. 17th Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, volume 629, pages 383–391, 1992; To appear in Theoretical Computer Science.Google Scholar
  8. 8.
    C. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes. J. Comput. System Sci., 43(3):425–440, 1991.Google Scholar
  9. 9.
    C. H. Papadimitriou. Computational Complexity. Addison-Wesley Publishing Company, 1994.Google Scholar
  10. 10.
    V. Raghavan. Bounded degree graph inference from walks. J. Comput. System Sci., 49:108–132, 1994.Google Scholar
  11. 11.
    S. Rudich. Inferring the structure of a Markov chain from its output. In Proc. 26th IEEE Symp. Foundations of Computer Science, pages 321–326, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  1. 1.Department of Information SystemsKyushu University 39KasugaJapan
  2. 2.Research Institute of Fundamental Information ScienceKyushu University 33FukuokaJapan

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