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2 -Way vs.d -Way Branching for CSP

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Principles and Practice of Constraint Programming - CP 2005 (CP 2005)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 3709))

Abstract

Most CSP algorithms are based on refinements and extensions of backtracking, and employ one of two simple “branching schemes”: 2-way branching or d-way branching, for domain size d. The schemes are not equivalent, but little is known about their relative power. Here we compare them in terms of how efficiently they can refute an unsatisfiable instance with optimal branching choices, by studying two variants of the resolution proof system, denoted C − RES and NG − RES, which model the reasoning of CSP algorithms. The tree-like restrictions, tree − C − RES and tree − NG − RES, exactly capture the power of backtracking with 2-way branching and d-way branching, respectively. We give a family instances which require exponential sized search trees for backtracking with d-way branching, but have size O(d 2 n) search trees for backtracking with 2-way branching. We also give a natural branching strategy with which backtracking with 2-way branching finds refutations of these instances in time O(d 2 n 2). The unrestricted variants of C − RES and NG − RES can simulate the reasoning of algorithms which incorporate learning and k-consistency enforcement. We show exponential separations between C − RES and NG − RES, as well as between the tree-like and unrestricted versions of each system. All separations given are nearly optimal.

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References

  1. Baker, A.B.: Intelligent Backtracking on Constraint Satisfaction Problems: Experimental and Theoretical Results. PhD thesis, University of Oregon (1995)

    Google Scholar 

  2. Beame, P., Kautz, H., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. Journal of Artificial Intelligence Research 22, 319–351 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Ben-Sasson, E., Impagliazzo, R., Wigderson, A.: Near-optimal separation of treelike and general resolution. Technical Report TR01-005, Electronic Colloquium on Computational Complexity (ECCC) (2000)

    Google Scholar 

  4. Bonet, M.L., Esteban, J.L., Galesi, N., Johansen, J.: Exponential separations between restricted resolution and cutting planes proof systems. In: Proc. of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1998), pp. 638–647. IEEE Press, Los Alamitos (1998)

    Google Scholar 

  5. Bonet, M.L., Galesi, N.: A study of proof search algorithms for resolution and polynomial calculus. In: Proc. 40th Symposium on Foundations of Computer Science, pp. 422–432 (1999)

    Google Scholar 

  6. Buresh-Oppenheim, J., Mitchell, D., Pitassi, T.: Linear and negative resolution are weaker than resolution. Technical Report TR01-074, Electronic Colloquium on Computational Complexity (ECCC) (2001)

    Google Scholar 

  7. Celoni, J.R., Paul, W.J., Tarjan, R.E.: Space bounds for a game on graphs. Mathematical Systems Theory 10, 239–251 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Joey Hwang, C.Y.: A theoretical comparison of resolution proof systems for csp algorithms. Master’s thesis, Simon Fraser University (2004)

    Google Scholar 

  9. Krishnamurthy, B.: Short proofs for tricky formulas. Acta Informatica 22, 253–274 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mitchell, D.G.: The Resolution Complexity of Constraint Satisfaction. PhD thesis, University of Toronto (2002)

    Google Scholar 

  11. Mitchell, D.G.: Resolution and constraint satisfaction. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 555–569. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  12. Park, V.: An empirical study of different branching strategies for constraint satisfaction problems. Master’s thesis, University of Waterloo (2004)

    Google Scholar 

  13. Smith, B.M., Sturdy, P.: An empirical investigation of value ordering for finding all solutions. Presented at the ECAI 2004 workshop on Modelling and Solving Problems with Constraints (2004)

    Google Scholar 

  14. Stalmarck, G.: Short resolution proofs for a sequence of tricky formulas. Acta Informatica 33, 277–280 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Computing Science, Simon Fraser University, Burnaby, V5A 1S6, Canada

    Joey Hwang & David G. Mitchell

Authors
  1. Joey Hwang
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  2. David G. Mitchell
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Editors and Affiliations

  1. Cheriton School of Computer Science, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    Peter van Beek

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© 2005 Springer-Verlag Berlin Heidelberg

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Hwang, J., Mitchell, D.G. (2005). 2 -Way vs.d -Way Branching for CSP. In: van Beek, P. (eds) Principles and Practice of Constraint Programming - CP 2005. CP 2005. Lecture Notes in Computer Science, vol 3709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11564751_27

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  • DOI: https://doi.org/10.1007/11564751_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29238-8

  • Online ISBN: 978-3-540-32050-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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