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Variable ordering for decision diagrams: A portfolio approach

Abstract

Relaxed decision diagrams have been successfully applied to solve combinatorial optimization problems, but their performance is known to strongly depend on the variable ordering. We propose a portfolio approach to selecting the best ordering among a set of alternatives. We consider several different portfolio mechanisms: a static uniform time-sharing portfolio, an offline predictive model of the single best algorithm using classifiers, a low-knowledge algorithm selection, and a dynamic online time allocator. As a case study, we compare and contrast their performance on the graph coloring problem. We find that on this problem domain, the dynamic online time allocator provides the best overall performance.

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Notes

  1. The code has been downloaded from https://github.com/heldstephan/exactcolors.

References

  1. Beck, J.C. & Freuder, E.C. (2004). Simple rules for low-knowledge algorithm selection. In: International conference on integration of artificial intelligence (AI) and operations research (OR) techniques in constraint programming (pp. 50–64). Springer

  2. Bergman, D., Cire, A.A., van Hoeve, W.J., Hooker, J.N. (2012). Variable ordering for the application of BDDs to the maximum independent set problem. In: Proceedings of CPAIOR. LNCS, vol. 298, (pp. 34–49). Springer

  3. Bergman, D., Cire, A. A., van Hoeve, W. J., & Hooker, J. N. (2014). Optimization bounds from binary decision diagrams. INFORMS Journal on Computing, 26(2), 253–268.

    MathSciNet  Article  Google Scholar 

  4. Bergman, D., Cire, A.A., van Hoeve, W.J., & Hooker, J.N. (2016). Decision diagrams for optimization. Springer.

  5. Bergman, D., Cire, A.A., Van Hoeve, W.J., Hooker, J.N. (2012). Variable ordering for the application of bdds to the maximum independent set problem. In: International conference on integration of artificial intelligence (AI) and operations research (OR) techniques in constraint programming (pp. 34–49). Springer

  6. Bouckaert, R.R., Frank, E., Hall, M., Kirkby, R., Reutemann, P., Seewald, A., & Scuse, D. (2016). Weka manual for version 3-9-1. Hamilton, New Zealand: University of Waikato.

  7. Brélaz, D. (1979). New methods to color the vertices of a graph. Communications of the ACM, 22(4), 251–256.

    MathSciNet  Article  Google Scholar 

  8. Bryant, R. E. (1986). Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers, C–35(8), 677–691.

    Article  Google Scholar 

  9. Bryant, R. E. (1992). Symbolic boolean manipulation with ordered binary decision diagrams. ACM Computing Surveys, 24, 293–318.

    Article  Google Scholar 

  10. Cappart, Q., Goutierre, E., Bergman, D., Rousseau, L.M. (2019) Improving optimization bounds using machine learning: Decision diagrams meet deep reinforcement learning. In: Proceedings of AAAI (pp. 1443–1451). AAAI Press

  11. Culberson, J. C., & Luo, F. (1996). Exploring the k-colorable landscape with iterated greedy. Cliques, coloring, and satisfiability: second DIMACS implementation challenge, 26, 245–284.

    MathSciNet  Article  Google Scholar 

  12. Freuder, E. C. (1982). A sufficient condition for backtrack-free search. Journal of the ACM (JACM), 29(1), 24–32.

    MathSciNet  Article  Google Scholar 

  13. Gagliolo, M., & Schmidhuber, J. (2011). Algorithm portfolio selection as a bandit problem with unbounded losses. Annals of Mathematics and Artificial Intelligence, 61(2), 49–86.

    MathSciNet  Article  Google Scholar 

  14. Gomes, C. P., & Selman, B. (2001). Algorithm portfolios. Artificial Intelligence, 126(1–2), 43–62.

    MathSciNet  Article  Google Scholar 

  15. Held, S., Cook, W., & Sewell, E. C. (2012). Maximum-weight stable sets and safe lower bounds for graph coloring. Mathematical Programming Computation, 4(4), 363–381.

    MathSciNet  Article  Google Scholar 

  16. van Hoeve, W.J. (2020). Graph coloring lower bounds from decision diagrams. In: Proceedings of IPCO. LNCS, vol. 12125, (pp. 405–419). Springer

  17. van Hoeve, W.J. (2021) Graph coloring with decision diagrams. Mathematical Programming, 1–44

  18. Johnson, D.S., Trick, M.A. (1996). Cliques, coloring, and satisfiability: second DIMACS implementation challenge, October 11-13, 1993, vol. 26. American Mathematical Soc

  19. Kotthoff, L. (2016). Algorithm selection for combinatorial search problems: A survey. In: Data mining and constraint programming, (pp. 149–190). Springer

  20. Musliu, N., Schwengerer, M. (2013). Algorithm selection for the graph coloring problem. In: International conference on learning and intelligent optimization. (pp. 389–403). Springer

  21. Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of small-worldnetworks. Nature, 393(6684), 440–442.

    Article  Google Scholar 

  22. Wegener, I. (2000). Branching programs and binary decision diagrams: Theory and applications. SIAM monographs on discrete mathematics and applications: Society for Industrial and Applied Mathematics.

  23. Xu, L., Hutter, F., Hoos, H. H., & Leyton-Brown, K. (2008). Satzilla: portfolio-based algorithm selection for sat. Journal of artificial intelligence research, 32, 565–606.

    Article  Google Scholar 

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Correspondence to Anthony Karahalios.

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Partially supported by Office of Naval Research Grants No. N00014-18-1-2129 and N00014-21-1-2240 and National Science Foundation Award #1918102.

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Karahalios, A., van Hoeve, WJ. Variable ordering for decision diagrams: A portfolio approach. Constraints 27, 116–133 (2022). https://doi.org/10.1007/s10601-021-09325-6

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Keywords

  • Decision diagrams
  • Graph coloring
  • Variable ordering