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Online Interval Coloring

1981; Kierstead, Trotter

  • Reference work entry
Encyclopedia of Algorithms

  • 362 Accesses

Keywords and Synonyms

An extremal problem in recursive combinatorics      

Problem Definition

Online interval coloring is a graph coloring problem. In such problems the vertices of a graph are presented one by one. Each vertex is presented in turn, along with a list of its edges in the graph, which are incident to previously presented vertices. The goal is to assign colors (which without loss of generality are assumed to be non-negative integers) to the vertices, so that two vertices which share an edge receive different colors, and the total number of colors used (or alternatively, the largest index of any color that is used) is minimized. The smallest number of colors, for which the graph still admits a valid coloring, is called the chromatic number of the graph.

The interval coloring problem is defined as follows. Intervals on the real line are presented one by one, and the online algorithm must assign each interval a color before the next interval arrives, so that no two intersecting...

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Notes

  1. 1.

    A graph G is perfect if any induced subgraph of G, G′ (including G), can be colored using ω(G′) colors, where ω(G′) is the size of the largest cardinality clique in G′. (For any graph, ω is a clear lower bound on its chromatic number).

Recommended Reading

  1. Adamy, U., Erlebach, T.: Online coloring of intervals with bandwidth. In: Proc. of the First International Workshop on Approximation and Online Algorithms (WAOA2003), pp. 1–12 (2003)

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  3. Bar-Noy, A., Motwani, R., Naor, J.: The greedy algorithm is optimal for on-line edge coloring. Inf. Proc. Lett. 44(5), 251–253 (1992)

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  4. Chrobak, M., Ślusarek, M.: On some packing problems relating to dynamical storage allocation. RAIRO J. Inf. Theor. Appl. 22, 487–499 (1988)

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  5. Epstein, L., Levin, A.: On the max coloring problem. In: Proc. of the Fifth International Workshop on Approximation and Online Algorithms (WAOA2007) (2007), pp. 142–155

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  6. Epstein, L., Levin, A., Woeginger, G.J.: Graph coloring with rejection. In: Proc. of 14th European Symposium on Algorithms (ESA2006), pp. 364–375. (2006)

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  7. Epstein, L., Levy, M.: Online interval coloring and variants. In: Proc. of The 32nd International Colloquium on Automata, Languages and Programming (ICALP2005), pp. 602–613. (2005)

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  8. Epstein, L., Levy, M.: Online interval coloring with packing constraints. In: Proc. of the 30th International Symposium on Mathematical Foundations of Computer Science (MFCS2005), pp. 295–307. (2005)

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  9. Gyárfás, A., Lehel, J.: Effective on-line coloring of P 5-free graphs. Combinatorica 11(2), 181–184 (1991)

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  10. Kierstead, H.A.: The linearity of first-fit coloring of interval graphs. SIAM J. Discret. Math. 1(4), 526–530 (1988)

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  11. Kierstead, H.A., Trotter, W.T.: An extremal problem in recursive combinatorics. Congr. Numerantium 33, 143–153 (1981)

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  12. Leonardi, S., Vitaletti, A.: Randomized lower bounds for online path coloring. In: Proc. of the second International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM'98), pp. 232–247. (1998)

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  13. Pemmaraju, S., Raman, R., Varadarajan, K.: Buffer minimization using max-coloring. In: Proc. of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 562–571. (2004)

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  14. Trotter, W.T.: Current research problems: First Fit colorings of interval graphs. http://www.math.gatech.edu/~trotter/rprob.htm Access date: December 24, 2007.

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Author information

Authors and Affiliations

  1. Department of Math, University of Haifa, Haifa, Israel

    Leah Epstein

Authors
  1. Leah Epstein
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Epstein, L. (2008). Online Interval Coloring. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_264

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