Advertisement

Complexity and Online Algorithms for Minimum Skyline Coloring of Intervals

  • Thomas Erlebach
  • Fu-Hong Liu
  • Hsiang-Hsuan Liu
  • Mordechai Shalom
  • Prudence W. H. Wong
  • Shmuel Zaks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10628)

Abstract

Graph coloring has been studied extensively in the literature. The classical problem concerns the number of colors used. In this paper, we focus on coloring intervals where the input is a set of intervals and two overlapping intervals cannot be assigned the same color. In particular, we are interested in the setting where there is an increasing cost associated with using a higher color index. Given a set of intervals (on a line) and a coloring, the cost of the coloring at any point is the cost of the maximum color index used at that point and the cost of the overall coloring is the integral of the cost over all points on the line. The objective is to assign a valid color to each interval and minimize the total cost of the coloring. Intuitively, the maximum color index used at each point forms a skyline and so the objective is to obtain a minimum skyline coloring. The problem arises in various applications including optical networks and job scheduling.

Alicherry and Bhatia defined in 2003 a more general problem in which the colors are partitioned into classes and the cost of a color depends solely on its class. This problem is NP-hard and the reduction relies on the fact that some color class has more than one color. In this paper we show that when each color class only contains one color, this simpler setting remains NP-hard via a reduction from the arc coloring problem. In addition, we initiate the study of the online setting and present an asymptotically optimal online algorithm. We further study a variant of the problem in which the intervals are already partitioned into sets and the objective is to assign a color to each set such that the total cost is minimum. We show that this seemingly easier problem remains NP-hard by a reduction from the optimal linear arrangement problem.

References

  1. 1.
    Adamy, U., Erlebach, T.: Online coloring of intervals with bandwidth. In: Solis-Oba, R., Jansen, K. (eds.) WAOA 2003. LNCS, vol. 2909, pp. 1–12. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24592-6_1 CrossRefGoogle Scholar
  2. 2.
    Alicherry, M., Bhatia, R.: Line system design and a generalized coloring problem. In: Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 19–30. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-39658-1_5 CrossRefGoogle Scholar
  3. 3.
    Azar, Y., Fiat, A., Levy, M., Narayanaswamy, N.S.: An improved algorithm for online coloring of intervals with bandwidth. Theor. Comput. Sci. 363(1), 18–27 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, New York (1998)zbMATHGoogle Scholar
  5. 5.
    Chang, J., Gabow, H.N., Khuller, S.: A model for minimizing active processor time. Algorithmica 70(3), 368–405 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chang, J., Khuller, S., Mukherjee. K.: LP rounding and combinatorial algorithms for minimizing active and busy time. In: SPAA, pp. 118–127 (2014)Google Scholar
  7. 7.
    Chrobak, M., Slusarek, M.: On some packing problem related to dynamic storage allocation. ITA 22(4), 487–499 (1988)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Flammini, M., Monaco, G., Moscardelli, L., Shachnai, H., Shalom, M., Tamir, T., Zaks, S.: Minimizing total busy time in parallel scheduling with application to optical networks. Theor. Comput. Sci. 411(40–42), 3553–3562 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Garey, M., Johnson, D.S., Miller, G., Papadimitriou, C.H.: The complexity of coloring circular arcs and chords. SIAM J. Algebraic Discrete Meth. 1(2), 216–227 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Halldórsson, M.M., Kortsarz, G., Shachnai, H.: Minimizing average completion of dedicated tasks and interval graphs. In: Goemans, M., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX/RANDOM -2001. LNCS, vol. 2129, pp. 114–126. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44666-4_15 CrossRefGoogle Scholar
  12. 12.
    Jansen, K.: Approximation results for the optimum cost chromatic partition problem. J. Algorithms 34(1), 54–89 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Khandekar, R., Schieber, B., Shachnai, H., Tamir, T.: Minimizing busy time in multiple machine real-time scheduling. In: FSTTCS, pp. 169–180 (2010)Google Scholar
  14. 14.
    Kierstead, H., Trotter, W.: An extremal problem in recursive combinatorics. Congressus Numerantium 33, 143–153 (1981)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kroon, L.G., Sen, A., Deng, H., Roy, A.: The optimal cost chromatic partition problem for trees and interval graphs. In: d’Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds.) WG 1996. LNCS, vol. 1197, pp. 279–292. Springer, Heidelberg (1997).  https://doi.org/10.1007/3-540-62559-3_23 CrossRefGoogle Scholar
  16. 16.
    Kubale, M. (ed.): Graph Colorings. American Mathematical Society, Providence (2004)zbMATHGoogle Scholar
  17. 17.
    Kumar, V., Rudra, A.: Approximation algorithms for wavelength assignment. In: Sarukkai, S., Sen, S. (eds.) FSTTCS 2005. LNCS, vol. 3821, pp. 152–163. Springer, Heidelberg (2005).  https://doi.org/10.1007/11590156_12 CrossRefGoogle Scholar
  18. 18.
    Mertzios, G., Shalom, M., Voloshin, A., Wong, P., Zaks, S.: Optimizing busy time on parallel machines. Theor. Comput. Sci. 562, 524–541 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Nicoloso, S., Sarrafzadeh, M., Song, X.: On the sum coloring problem on interval graphs. Algorithmica 23(2), 109–126 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Pemmaraju, S.V., Raman, R., Varadarajan, K.R.: Buffer minimization using max-coloring. In: SODA, pp. 562–571 (2004)Google Scholar
  21. 21.
    Ren, R., Tang, X.: Clairvoyant dynamic bin packing for job scheduling with minimum server usage time. In: SPAA, pp. 227–237 (2016)Google Scholar
  22. 22.
    Shalom, M., Voloshin, A., Wong, P., Yung, F., Zaks, S.: Online optimization of busy time on parallel machines. Theor. Comput. Sci. 560, 190–206 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Winkler, P., Zhang, L.: Wavelength assignment and generalized interval graph coloring. In: SODA, pp. 830–831 (2003)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Thomas Erlebach
    • 1
  • Fu-Hong Liu
    • 2
  • Hsiang-Hsuan Liu
    • 3
  • Mordechai Shalom
    • 4
  • Prudence W. H. Wong
    • 5
  • Shmuel Zaks
    • 6
  1. 1.Department of InformaticsUniversity of LeicesterLeicesterUK
  2. 2.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan
  3. 3.Institute of Computer ScienceUniversity of WrocławWrocławPoland
  4. 4.TelHai CollegeUpper GalileeIsrael
  5. 5.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  6. 6.Department of Computer ScienceTechnionHaifaIsrael

Personalised recommendations