Capacitated max -Batching with Interval Graph Compatibilities

  • Tim Nonner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)


We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the weight of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving a long-standing open problem, we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.


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  1. 1.
    Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baptiste, P.: Scheduling unit tasks to minimize the number of idle periods: a polynomial time algorithm for offline dynamic power management. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), pp. 364–367 (2006)Google Scholar
  3. 3.
    Becchetti, L., Korteweg, P., Marchetti-Spaccamela, A., Skutella, M., Stougie, L., Vitaletti, A.: Latency constrained aggregation in sensor networks. In: Proceedings of the 14th Annual European Symposium on Algorithms (ESA’06), pp. 88–99 (2006)Google Scholar
  4. 4.
    Bodlaender, H.L., Jansen, K.: Restrictions of graph partition problems. part I. Theoretical Computer Science 148(1), 93–109 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Boudhar, M.: Dynamic scheduling on a single batch processing machine with split compatibility graphs. Journal of Mathematical Modelling and Algorithms 2(1), 17–35 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Correa, J., Megow, N., Raman, R., Suchan, K.: Cardinality constrained graph partitioning into cliques with submodular costs. In: Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW’09), pp. 347–350 (2009)Google Scholar
  7. 7.
    Even, G., Levi, R., Rawitz, D., Schieber, B., Shahar, S., Sviridenko, M.: Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs. ACM Transactions on Algorithms 4(3) (2008)Google Scholar
  8. 8.
    Finke, G., Jost, V., Queyranne, M., Sebö, A.: Batch processing with interval graph compatibilities between tasks. Discrete Applied Mathematics 156(5), 556–568 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gijswijt, D., Jost, V., Queyranne, M.: Clique partitioning of interval graphs with submodular costs on the cliques. RAIRO - Operations Research 41(3), 275–287 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Hochbaum, D.S., Laundy, D.: Scheduling semiconductor burn-in operations to minimize total flowtime. Operations Research 45, 874–885 (1997)zbMATHCrossRefGoogle Scholar
  11. 11.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM 32(1), 130–136 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intracability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)Google Scholar
  13. 13.
    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), pp. 1064–1075 (2005)Google Scholar
  14. 14.
    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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tim Nonner
    • 1
  1. 1.Albert Ludwigs University of FreiburgGermany

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