On the proper intervalization of colored caterpillar trees

Carme Àlvarez; Maria Serna

RAIRO - Theoretical Informatics and Applications (2009)

  • Volume: 43, Issue: 4, page 667-686
  • ISSN: 0988-3754

Abstract

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This paper studies the computational complexity of the proper interval colored graph problem (PICG), when the input graph is a colored caterpillar, parameterized by hair length. In order prove our result we establish a close relationship between the PICG and a graph layout problem the proper colored layout problem (PCLP). We show a dichotomy: the PICG and the PCLP are NP-complete for colored caterpillars of hair length ≥2, while both problems are in P for colored caterpillars of hair length <2. For the hardness results we provide a reduction from the multiprocessor scheduling problem, while the polynomial time results follow from a characterization in terms of forbidden subgraphs.

How to cite

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Àlvarez, Carme, and Serna, Maria. "On the proper intervalization of colored caterpillar trees." RAIRO - Theoretical Informatics and Applications 43.4 (2009): 667-686. <http://eudml.org/doc/250603>.

@article{Àlvarez2009,
abstract = { This paper studies the computational complexity of the proper interval colored graph problem (PICG), when the input graph is a colored caterpillar, parameterized by hair length. In order prove our result we establish a close relationship between the PICG and a graph layout problem the proper colored layout problem (PCLP). We show a dichotomy: the PICG and the PCLP are NP-complete for colored caterpillars of hair length ≥2, while both problems are in P for colored caterpillars of hair length <2. For the hardness results we provide a reduction from the multiprocessor scheduling problem, while the polynomial time results follow from a characterization in terms of forbidden subgraphs. },
author = {Àlvarez, Carme, Serna, Maria},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Complexity; caterpillar tree; graph layout problems; coloring; complexity},
language = {eng},
month = {7},
number = {4},
pages = {667-686},
publisher = {EDP Sciences},
title = {On the proper intervalization of colored caterpillar trees},
url = {http://eudml.org/doc/250603},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Àlvarez, Carme
AU - Serna, Maria
TI - On the proper intervalization of colored caterpillar trees
JO - RAIRO - Theoretical Informatics and Applications
DA - 2009/7//
PB - EDP Sciences
VL - 43
IS - 4
SP - 667
EP - 686
AB - This paper studies the computational complexity of the proper interval colored graph problem (PICG), when the input graph is a colored caterpillar, parameterized by hair length. In order prove our result we establish a close relationship between the PICG and a graph layout problem the proper colored layout problem (PCLP). We show a dichotomy: the PICG and the PCLP are NP-complete for colored caterpillars of hair length ≥2, while both problems are in P for colored caterpillars of hair length <2. For the hardness results we provide a reduction from the multiprocessor scheduling problem, while the polynomial time results follow from a characterization in terms of forbidden subgraphs.
LA - eng
KW - Complexity; caterpillar tree; graph layout problems; coloring; complexity
UR - http://eudml.org/doc/250603
ER -

References

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  1. C. Àlvarez, J. Díaz and M. Serna, The hardness of intervalizing four colored caterpillars. Discrete Math.235 (2001) 19–27.  Zbl0977.05055
  2. C. Àlvarez, J. Díaz and M. Serna, Intervalizing colored graphs is NP-complete for caterpillars with hair length 2. Technical Report LSI 98-9-R, Universitat Politècnica de Catalunya (1998).  
  3. H. Bodlaender, M.R. Fellows and M.T. Hallet, Beyond NP-completeness for problems of bounded width: hardness for the W-hierarchy, in 26th ACM Symposium on Theory of Computing (1994) 449–458.  Zbl1345.68152
  4. J. Díaz, A.M. Gibbons, M.S. Paterson and J. Torán, The minsumcut problem, in Algorithms and Datastructure, edited by F. Dehen, R.J. Sack and N. Santoro. Lect. Notes Comput. Sci.519 (1991) 65–79.  Zbl0764.68064
  5. M.J. Dinneen, VLSI Layouts and DNA physical mappings. Technical Report, Los Alamos National Laboratory (1996).  
  6. M.R. Fellows, M.T. Hallet and W.T. Wareham, DNA physical mapping: Three ways difficult, in Algorithms-ESA'93, edited by T. Lengauer. Lect. Notes Comput. Sci.726 (1993) 157–168.  
  7. P.W. Goldberg, M.C. Golumbic, H. Kaplan and R. Shamir, Four strikes against physical mapping of DNA. J. Comput. Biol.2 (1995) 139–152.  
  8. M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979).  Zbl0411.68039
  9. M.C. Golumbic, H. Kaplan and R. Shamir, On the complexity of DNA physical mapping. Adv. Appl. Math.15 (1994) 203–215.  Zbl0806.92007
  10. M.C. Golumbic, H. Kaplan and R. Shamir, Graph sandwich problems. J. Algorithms19 (1995) 449–473.  Zbl0838.68054
  11. M.C. Golumbic, Algorithmic graph theory and perfect graphs. Academic Press, New York (1980).  Zbl0541.05054
  12. M.C. Golumbic and R. Shamir, Complexity and algorithms for reasoning about time: A graph theoretical approach. J. ACM40 (1993) 1108–1113.  Zbl0795.68095
  13. D. Kuo and G.J. Chang, The profile minimization problem in trees. SIAM J. Comput.23 (1994) 71–81.  Zbl0794.05117
  14. H. Kaplan and R. Shamir, Pathwidth, bandwidth and completion problems to proper interval graphs with small cliques. SIAM J. Comput.25 (1996) 540–561.  Zbl0852.68072
  15. H. Kaplan, R. Shamir and R.E. Tarjan, Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput.28 (1999) 1906-1922.  Zbl0928.68124
  16. B. Monien, The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Algebr. Discrete Methods7 (1986) 505–512.  Zbl0624.68059

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