Partitioning Based Algorithms for Some Colouring Problems

  • Ola Angelsmark
  • Johan Thapper
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3978)


We discuss four variants of the graph colouring problem, and empresent algorithms for solving them. The problems are k -Colourability, Max Ind k -COL, Max Val k -COL, and, finally, Max k -COL, which is the unweighted case of the Max k -Cut problem. The algorithms are based on the idea of partitioning the domain of the problems into disjoint subsets, and then considering all possible instances were the variables are restricted to values from these partitions. If a pair of variables have been restricted to different partitions, then the constraint between them is always satisfied since the only allowed constraint is disequality.


Domain Size Chromatic Number Colouring Problem Register Allocation Graph Colouring Problem 
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  1. 1.
    Angelsmark, O.: Constructing Algorithms for Constraint Satisfaction and Related Problems. PhD thesis, Department of Computer and Information Science, Linköpings Universitet, Sweden (2005)Google Scholar
  2. 2.
    Angelsmark, O., Jonsson, P.: Improved Algorithms for Counting Solutions in Constraint Satisfaction Problems. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 81–95. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Angelsmark, O., Jonsson, P., Thapper, J.: Two methods for constructing new CSP algorithms from old. Unpublished manuscript (2004)Google Scholar
  4. 4.
    Angelsmark, O., Thapper, J.: A microstructure based approach to constraint satisfaction optimisation problems. In: Russell, I., Markov, Z. (eds.) Recent Advances in Artificial Intelligience. Proceedings of the Eighteenth International Florida Artificial Intelligence Research Society Conference (FLAIRS-2005), Clearwater Beach, Florida, USA, May 15-17, 2005, pp. 155–160. AAAI Press, Menlo Park (2005)Google Scholar
  5. 5.
    Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Operations Research Letters 32(6), 547–556 (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Byskov, J.M.: Exact Algorithms for Graph Colouring and Exact Satisfiability. PhD thesis, Basic Research In Computer Science (BRICS), Department of Computer Science, University of Aarhus, Denmark (August 2004)Google Scholar
  7. 7.
    Byskov, J.M., Eppstein, D.: An algorithm for enumerating maximal bipartite subgraphs. Unpublished manuscript (see also [6]) (2004)Google Scholar
  8. 8.
    Chaitin, G.J., Auslander, M.A., Chandra, A.K., Cocke, J., Hopkins, M.E., Markstein, P.W.: Register allocation via coloring. Computer Languages 6, 47–57 (1981)CrossRefGoogle Scholar
  9. 9.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280 (1990)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2SAT and 3SAT formulae. Theoretical Computer Science 332(1–3), 265–291 (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Eppstein, D.: Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2001), Washington, DC, USA, January 7-9, 2001, pp. 329–337. ACM/SIAM (2001)Google Scholar
  12. 12.
    Eppstein, D.: Small maximal independent sets and faster exact graph coloring. Journal of Graph Algorithms and Applications 7(2), 131–140 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Feder, T., Motwani, R.: Worst-case time bounds for coloring and satisfiability problems. Journal of Algorithms 45(2), 192–201 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gamst, A.: Some lower bounds for a class of frequency assignment problems. IEEE Transactions on Vehicular Technology 35(1), 8–14 (1986)CrossRefGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  16. 16.
    Jégou, P.: Decomposition of domains based on the micro-structure of finite constraint-satisfaction problems. In: Proceedings of the 11th (US) National Conference on Artificial Intelligence (AAAI-1993), Washington DC, USA, July 1993, pp. 731–736. American Association for Artificial Intelligence (AAAI), Menlo Park (1993)Google Scholar
  17. 17.
    Jonsson, P., Liberatore, P.: On the complexity of finding satisfiable subinstances in constraint satisfaction. Technical Report TR99-038, Electronic Colloquium on Computational Complexity (1999)Google Scholar
  18. 18.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)Google Scholar
  19. 19.
    Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science 223(1–2), 1–72 (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Lawler, E.L.: A note on the complexity of the chromatic number problem. Information Processing Letters 5(3), 66–67 (1976)CrossRefzbMATHGoogle Scholar
  21. 21.
    Robson, M.: Finding a maximum independent set in time O(2n/4). Technical report, LaBRI, Université Bordeaux I (2001)Google Scholar
  22. 22.
    Williams, R.: A New Algorithm for Optimal Constraint Satisfaction and Its Implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ola Angelsmark
    • 1
  • Johan Thapper
    • 2
  1. 1.Department of Computer ScienceLund UniversityLundSweden
  2. 2.Department of MathematicsLinköpings UniversitetLinköpingSweden

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