Abstract
The heavy-tailed phenomenon that characterises the runtime distributions of backtrack search procedures has received considerable attention over the past few years. Some have conjectured that heavy-tailed behaviour is largely due to the characteristics of the algorithm used. Others have conjectured that problem structure is a significant contributor. In this paper we attempt to explore the former hypothesis, namely we study how variable and value ordering heuristics impact the heavy-tailedness of runtime distributions of backtrack search procedures. We demonstrate that heavy-tailed behaviour can be eliminated from particular classes of random problems by carefully selecting the search heuristics, even when using chronological backtrack search. We also show that combinations of good search heuristics can eliminate heavy tails from Quasigroups with Holes of order 10, and give some insights into why this is the case. These results motivate a more detailed analysis of the effects that variable and value ordering can have on heavy-tailedness. We show how combinations of variable and value ordering heuristics can result in a runtime distribution being inherently heavy-tailed. Specifically, we show that even if we were to use an Oracle to refute insoluble subtrees optimally, for some combinations of heuristics we would still observe heavy-tailed behaviour. Finally, we study the distributions of refutation sizes found using different combinations of heuristics and gain some further insights into what characteristics tend to give rise to heavy-tailed behaviour.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Achlioptas, D., Gomes, C.P., Kautz, H.A., Selman, B.: Generating satisfiable problem instances. In: Proceedings of AAAI 2000, pp. 256–261 (2000)
Bessière, C., Fernández, C., Gomes, C.P., Valls, M.: Pareto-like distributions in random binary csp. In: Proceedings of ACIA 2003 (2004)
Bessière, C., Regin, J.-C.: MAC and combined heuristics: two reasons to forsake FC (and CBJ?) on hard problems. In: Freuder, E.C. (ed.) CP 1996. LNCS, vol. 1118, pp. 61–75. Springer, Heidelberg (1996)
Boussemart, F., Hemery, F., Lecoutre, C., Sais, L.: Boosting systematic search by weighting constraints. In: Proceedings of ECAI 2004, pp. 146–150 (2004)
Brélaz, D.: New methods to color the vertices of a graph. Communications of the ACM 22(4), 251–256 (1979)
Colbourn, C.: Embedding partial steiner triple systems is NP-complete. Combinatorial Theory A(35), 100–105 (1983)
Gent, I.P., Walsh, T.: Easy problems are sometimes hard. Artificial Intelligence 70, 335–345 (1994)
Gomes, C.P., Fernández, C., Selman, B., Bessière, C.: Statistical regimes across constrainedness regions. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 32–46. Springer, Heidelberg (2004)
Gomes, C.P., Selman, B., Crato, N.: Heavy-tailed distributions in combinatorial search. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 121–135. Springer, Heidelberg (1997)
Gomes, C.P., Selman, B., Crato, N., Kautz, H.: Heavy-tailed phenomena in satisfiability and constraint satisfaction problems. Automated Reasoning 24(1/2), 67–100 (2000)
Haralick, R.M., Elliott, G.L.: Increasing tree search efficiency for constraint satisfaction problems. Artificial Intelligence 14(3), 263–313 (1980)
Hogg, T., Williams, C.P.: The hardest constraint problems: A double phase transition. Artificial Intelligence 69, 359–377 (1994)
Hulubei, T., O’Sullivan, B.: Optimal refutations for constraint satisfaction problems. In: Proceedings of IJCAI 2005 (2005)
Jacobson, M.T., Matthews, P.: Generating uniformly distributed random latin squares. Combinatorial Design 4, 405–437 (1996)
Kautz, H.A., Ruan, Y., Achlioptas, D., Gomes, C.P., Selman, B., Stickel, M.E.: Balance and filtering in structured satisfiable problems. In: Proceedings of IJCAI 2001, pp. 351–358 (2001)
Korf, R.E.: Depth-first iterative-deepening: an optimal admissible tree search. Artificial Intelligence 27(1), 97–109 (1985)
Lecoutre, C., Boussemart, F., Hemery, F.: Backjump-based techniques versus conflict-directed heuristics. In: Proceedings of ICTAI 2004, pp. 549–557 (2004)
Mackworth, A.K.: Consistency in networks of relations. Artificial Intelligence 8(1), 99–118 (1977)
Sabin, D., Freuder, E.C.: Contradicting conventional wisdom in constraint satisfaction. In: Proceedings of ECAI 1994, pp. 125–129 (1994)
Smith, B.M.: In search of exceptionally difficult constraint satisfaction problems. In: Meyer, M. (ed.) Constraint Processing. LNCS, vol. 923, pp. 139–156. Springer, Heidelberg (1995)
Smith, B.M., Grant, S.: Sparse constraint graphs and exceptionally hard problems. In: Proceedings of IJCAI 1995, pp. 646–651 (1995)
Smith, B.M., Sturdy, P.: An empirical investigation of value ordering for finding all solutions. In: Workshop on Modelling and Solving Problems with Constraints (2004)
Walsh, T.: Search in a small world. In: Proceedings of IJCAI 1999, pp. 1172–1177 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hulubei, T., O’Sullivan, B. (2005). Search Heuristics and Heavy-Tailed Behaviour. 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_26
Download citation
DOI: https://doi.org/10.1007/11564751_26
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)