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Complexity of Splits Reconstruction for Low-Degree Trees

  • Serge Gaspers
  • Mathieu Liedloff
  • Maya Stein
  • Karol Suchan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)

Abstract

Given a vertex-weighted tree T, the split of an edge xy in T is min{s x , s y } where s x (respectively, s y ) is the sum of all weights of vertices that are closer to x than to y (respectively, closer to y than to x) in T. Given a set of weighted vertices V and a multiset of splits S, we consider the problem of constructing a tree on V whose splits correspond to S. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be no more than 4. We show that

  • the problem is strongly NP-complete when T is required to be a path. For this variant we exhibit an algorithm that runs in polynomial time when the number of distinct vertex weights is constant.We also show that

  • the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be no more than 3, and

  • it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3.

Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm.

The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong correlation between structural properties, such as the Wiener index, which is closely connected to the considered problem, and biological activity.

Keywords

Unit Weight Maximum Degree Dynamic Programming Algorithm Wiener Index Vertex Weight 
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.
    Aoki-Kinoshita, K.F., Kanehisa, M., Kao, M.-Y., Li, X.-Y., Wang, W.: A 6-Approximation Algorithm for Computing Smallest Common Aon-Supertree with Application to the Reconstruction of Glycan Trees. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 100–110. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Balaban, A.T.: Chemical Applications of Graph Theory. Academic Press, Inc. (1976)Google Scholar
  3. 3.
    Bonchev, D., Rouvray, D.H.: Chemical Graph Theory: Introduction and Fundamentals. Taylor & Francis (1991)Google Scholar
  4. 4.
    Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index of trees: Theory and applications. Acta Applicandae Mathematicae 66(3), 211–249 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Faulon, J.-L., Bender, A.: Handbook of Chemoinformatics Algorithms, 1st edn. Chapman and Hall/CRC (2010)Google Scholar
  7. 7.
    Fellows, M.R., Gaspers, S., Rosamond, F.A.: Parameterizing by the Number of Numbers. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 123–134. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)zbMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Gillet, V.J., Willett, P., Bradshawand, J., Green, D.V.S.: Selecting combinatorial libraries to optimize diversity and physical properties. Journal of Chemical Information and Computer Sciences 39(1), 169–177 (1999)CrossRefGoogle Scholar
  11. 11.
    Goldman, D., Istrail, S., Lancia, G., Piccolboni, A., Walenz, B.: Algorithmic strategies in combinatorial chemistry. In: SODA, pp. 275–284 (2000)Google Scholar
  12. 12.
    Hammer, P.L. (ed.): Special issue on the 50th anniversary of the Wiener index. Discrete Applied Mathematics, vol. 80. Elsevier (1997)Google Scholar
  13. 13.
    Hulett, H., Will, T.G., Woeginger, G.J.: Multigraph realizations of degree sequences: Maximization is easy, minimization is hard. Operations Research Letters 36(5), 594–596 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Deconstructing Intractability: A Case Study for Interval Constrained Coloring. In: Kucherov, G., Ukkonen, E. (eds.) CPM 2009 Lille. LNCS, vol. 5577, pp. 207–220. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Li, X., Zhang, X.: The edge split reconstruction problem for chemical trees is NP-complete. MATCH Communications in Mathematical and in Computer Chemistry 51, 205–210 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sheridan, R.P., Kearsley, S.K.: Using a genetic algorithm to suggest combinatorial libraries. Journal of Chemical Information and Computer Sciences 35(2), 310–320 (1995)Google Scholar
  18. 18.
    Trinajstić, N.: Chemical Graph Theory, 2nd edn. CRC Press (1992)Google Scholar
  19. 19.
    Wiener, H.: Structural determination of paraffin boiling points. Journal of the American Chemical Society 69(1), 17–20 (1947)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Serge Gaspers
    • 1
  • Mathieu Liedloff
    • 2
  • Maya Stein
    • 3
  • Karol Suchan
    • 4
    • 5
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria
  2. 2.LIFOUniversité d’OrléansOrléansFrance
  3. 3.CMMUniversidad de ChileSantiagoChile
  4. 4.FICUniversidad Adolfo IbáñezSantiagoChile
  5. 5.WMSAGH - University of Science and TechnologyKrakowPoland

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