On the proper intervalization of colored caterpillar trees

Àlvarez, Carme; Serna, Maria

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 4, pp. 667-686.

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 $\ge$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.

Classification : 68Q25, 68W10
Mots clés : complexity, caterpillar tree, graph layout problems, coloring

[1] C. Àlvarez, J. Díaz and M. Serna, The hardness of intervalizing four colored caterpillars. Discrete Math. 235 (2001) 19-27. | MR | Zbl

[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.

[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. | MR | Zbl

[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. | MR

[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). | MR | Zbl

[9] M.C. Golumbic, H. Kaplan and R. Shamir, On the complexity of DNA physical mapping. Adv. Appl. Math. 15 (1994) 203-215. | MR | Zbl

[10] M.C. Golumbic, H. Kaplan and R. Shamir, Graph sandwich problems. J. Algorithms 19 (1995) 449-473. | MR | Zbl

[11] M.C. Golumbic, Algorithmic graph theory and perfect graphs. Academic Press, New York (1980). | MR | Zbl

[12] M.C. Golumbic and R. Shamir, Complexity and algorithms for reasoning about time: A graph theoretical approach. J. ACM 40 (1993) 1108-1113. | MR | Zbl

[13] D. Kuo and G.J. Chang, The profile minimization problem in trees. SIAM J. Comput. 23 (1994) 71-81. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[16] B. Monien, The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Algebr. Discrete Methods 7 (1986) 505-512. | MR | Zbl