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Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

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We are given an interval graph \( G = (V,E) \) where each interval \( I \in V\) has a weight \( w_I \in \mathbb {Q}^+ \). The goal is to color the intervals \( V\) with an arbitrary number of color classes \( C_1, C_2, \ldots , C_k \) such that \( \sum _{i=1}^k \max _{I \in C_i} w_I \) is minimized. This problem, called max-coloring interval graphs or weighted coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard [21] and presented a 2-approximation algorithm. We settle the approximation complexity of this problem by giving a polynomial-time approximation scheme (PTAS), that is, we show that there is an \( (1+\epsilon ) \)-approximation algorithm for any \( \epsilon > 0 \). The PTAS also works for the bounded case where the sizes of the color classes are bounded by some arbitrary \( k \le n \).

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  1. 1.

    All logarithms have base 2.


  1. 1.

    Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)

  2. 2.

    Bampis, E., Kononov, A., Lucarelli, G., Milis, I.: Bounded max-colorings of graphs. J. Disc. Algorithms 26, 56–68 (2014)

  3. 3.

    Bansal, N., Chakrabarti, A., Epstein, A., Schieber, B.: A quasi-ptas for unsplittable flow on line graphs. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC’06), pp. 721–729, (2006)

  4. 4.

    Becchetti, L., Marchetti-Spaccamela, A., Vitaletti, A., Korteweg, P., Skutella, M., Stougie, L.: Latency-constrained aggregation in sensor networks. ACM Trans. Algorithms 6(1), 88–99 (2009)

  5. 5.

    Buchsbaum, A.L., Karloff, H.J., Kenyon, C., Reingold, N., Thorup, M.: OPT versus LOAD in dynamic storage allocation. SIAM J. Comput. 33(3), 632–646 (2004)

  6. 6.

    de Werra, D., Demange, M., Escoffier, B., Monnot, J., Paschos, V.T.: Weighted coloring on planar, bipartite and split graphs: complexity and approximation. Disc. Appl. Math. 157(4), 819–832 (2009)

  7. 7.

    Epstein, L., Levin, A.: On the max coloring problem. Theor. Comput. Sci. 462, 23–38 (2012)

  8. 8.

    Escoffier, B., Monnot, J., Paschos, V.T.: Weighted coloring: further complexity and approximability results. Inf. Process. Lett 97(3), 98–103 (2006)

  9. 9.

    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57(2), 187–199 (1998)

  10. 10.

    Finke, G., Jost, V., Queyranne, M., Sebö, A.: Batch processing with interval graph compatibilities between tasks. Disc. Appl. Math. 156(5), 556–568 (2008)

  11. 11.

    Gröschel, M., Lovász, László, Schrijver, Alexander: Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin (1988)

  12. 12.

    Guan, D.J., Xuding, Z.: A coloring problem for weighted graphs. Inf. Process. Lett 61(2), 77–81 (1997)

  13. 13.

    Halldórsson, M.M., Shachnai, H.: Batch coloring flat graphs and thin. In :Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (SWAT’08), pp. 198–209. Springer, Berlin (2008)

  14. 14.

    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)

  15. 15.

    Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22, 463–468 (1975)

  16. 16.

    Kavitha, T., Mestre, J.: Max-coloring paths: tight bounds and extensions. J. Comb. Optim. 24(1), 1–14 (2012)

  17. 17.

    Mestre, J., Raman, R.: Max-coloring. In :Handbook of Combinatorial Optimization, Springer, New York, pp. 1871–1911, (2013)

  18. 18.

    Nonner, T.: Capacitated max-batching with interval graph compatibilities. Theor. Comput. Sci. 613, 79–93 (2016)

  19. 19.

    Pemmaraju, S. V., Raman, R.: Approximation algorithms for the max-coloring problem. In :Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP’05), vol. 5, pp. 1064–1075, (2005)

  20. 20.

    Pemmaraju, S. V., Raman, R., Varadarajan, K.: Buffer minimization using max-coloring. In :Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’04), pp. 562–571, (2004)

  21. 21.

    Pemmaraju, S.V., Raman, R., Varadarajan, K.R.: Max-coloring and online coloring with bandwidths on interval graphs. ACM Trans. Algorithms 7(3), 35 (2011)

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Correspondence to Tim Nonner.

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A preliminary version of this paper has appeared at ICALP’11.

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Nonner, T. Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs. Algorithmica 80, 2941–2956 (2018).

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  • Combinatorial optimization
  • Approximation algorithms
  • Approximation schemes
  • Graph algorithms
  • Interval graphs