Abstract
We consider online coloring of intervals with bandwidth in a setting where colors have variable capacities. Whenever the algorithm opens a new color, it must choose the capacity for that color and cannot change it later. The goal is to minimize the total capacity of all the colors used. We consider the bounded model, where all capacities must be chosen in the range (0,1], and the unbounded model, where the algorithm may use colors of any positive capacity. For the absolute competitive ratio, we give an upper bound of 14 and a lower bound of 4.59 for the bounded model, and an upper bound of 4 and a matching lower bound of 4 for the unbounded model. We also consider the offline version of these problems and show that the unbounded model is polynomially solvable, while the bounded model is NP-hard in the strong sense and admits a 3.6-approximation algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Aspnes, J., Azar, Y., Fiat, A., Plotkin, S.A., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. Journal of the ACM 44(3), 486–504 (1997)
Baruah, S.K., Koren, G., Mao, D., Mishra, B., Raghunathan, A., Rosier, L.E., Shasha, D., Wang, F.: On the competitiveness of on-line real-time task scheduling. Real-Time Systems 4(2), 125–144 (1992)
Chrobak, M., Ślusarek, M.: On some packing problems relating to dynamical storage allocation. RAIRO Journal on Information Theory and Applications 22, 487–499 (1988)
Coffman, E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: A survey. In: Hochbaum, D. (ed.) Approximation algorithms. PWS Publishing Company (1997)
Csirik, J.: An online algorithm for variable-sized bin packing. Acta Informatica 26, 697–709 (1989)
Csirik, J., Woeginger, G.J.: On-line packing and covering problems. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms: The State of the Art. LNCS, vol. 1442, pp. 147–177. Springer, Heidelberg (1998)
Epstein, L., Levy, M.: Online interval coloring and variants. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 602–613. Springer, Heidelberg (2005)
Epstein, L., Levy, M.: Online interval coloring with packing constraints. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 295–307. Springer, Heidelberg (2005)
Friesen, D.K., Langston, M.A.: Variable sized bin packing. SIAM J. Comput. 15, 222–230 (1986)
Jensen, T.R., Toft, B.: Graph coloring problems. Wiley, Chichester (1995)
Kierstead, H.A.: The linearity of first-fit coloring of interval graphs. SIAM Journal on Discrete Mathematics 1(4), 526–530 (1988)
Kierstead, H.A., Qin, J.: Coloring interval graphs with First-Fit. SIAM Journal on Discrete Mathematics 8, 47–57 (1995)
Kierstead, H.A., Trotter, W.T.: An extremal problem in recursive combinatorics. Congressus Numerantium 33, 143–153 (1981)
Murgolo, F.D.: An efficient approximation scheme for variable-sized bin packing. SIAM J. Comput. 16(1), 149–161 (1987)
Narayanaswamy, N.S.: Dynamic storage allocation and online colouring interval graphs. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 329–338. Springer, Heidelberg (2004)
Pemmaraju, S.V., Raman, R., Varadarajan, K.R.: Buffer minimization using max-coloring. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 562–571 (2004)
Seiden, S.S.: An optimal online algorithm for bounded space variable-sized bin packing. SIAM Journal on Discrete Mathematics 14(4), 458–470 (2001)
Seiden, S.S.: On the online bin packing problem. Journal of the ACM 49(5), 640–671 (2002)
Seiden, S.S., van Stee, R., Epstein, L.: New bounds for variable-sized online bin packing. SIAM Journal on Computing 32(2), 455–469 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Epstein, L., Erlebach, T., Levin, A. (2006). Variable Sized Online Interval Coloring with Bandwidth. In: Arge, L., Freivalds, R. (eds) Algorithm Theory – SWAT 2006. SWAT 2006. Lecture Notes in Computer Science, vol 4059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785293_6
Download citation
DOI: https://doi.org/10.1007/11785293_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35753-7
Online ISBN: 978-3-540-35755-1
eBook Packages: Computer ScienceComputer Science (R0)