Abstract
Many studies have been conducted on the complexity of Constraint Satisfaction Problem (CSP) classes. However, there exists little theoretical work on the hardness of individual CSP instances. In this context, the backdoor key fraction (BKF) [17] was introduced as a quantifier of problem hardness for individual satisfiable instances with regard to backtracking search. In our paper, after highlighting the weaknesses of the BKF, we propose a better characterization of the hardness of an individual satisfiable CSP instance based on the ratio between the size of the solution space and that of the search space. We formally show that our measure is negatively correlated with instance hardness. We also show through experiments that this measure evaluates more accurately the hardness of individual instances than the BKF.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Achlioptas, D., Gomes, C.P., Kautz, H.A., Selman, B.: Generating satisfiable problem instances. In: Proceedings of AAAI, IAAI, Austin, Texas, USA, 30 July–3 August 2000, pp. 256–261 (2000)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)
Dechter, R.: Constraint Processing. Elsevier/Morgan Kaufmann, New York City/Burlington (2003)
Escamocher, G., O’Sullivan, B.: On the minimal constraint satisfaction problem: complexity and generation. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 731–745. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26626-8_54
Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29(1), 24–32 (1982)
Freuder, E.C.: Complexity of K-tree structured constraint satisfaction problems. In: Proceedings of AAAI, Boston, Massachusetts, 29 July–3 August 1990, vol. 2, pp. 4–9 (1990)
Freuder, E.C.: Completable representations of constraint satisfaction problems. In: Proceedings of KR, Cambridge, MA, USA, 22–25 April 1991, pp. 186–195 (1991)
Ganian, R., Ramanujan, M.S., Szeider, S.: Combining treewidth and backdoors for CSP. In: 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, Hannover, Germany, 8–11 March 2017, pp. 36:1–36:17 (2017)
Gent, I.P., MacIntyre, E., Prosser, P., Walsh, T.: The constrainedness of search. In: Proceedings of AAAI, IAAI, Portland, Oregon, 4–8 August 1996, vol. 1, pp. 246–252 (1996)
Gomes, C.P., Fernández, C., Selman, B., Bessière, C.: Statistical regimes across constrainedness regions. Constraints 10(4), 317–337 (2005)
Hebrard, E.: Mistral, a constraint satisfaction library. In: Proceedings of the Third International CSP Solver Competition, vol. 3, p. 3 (2008)
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, Seattle, Washington, USA, 4–10 August 2001, pp. 351–358 (2001)
Larrosa, J., Schiex, T.: Solving weighted CSP by maintaining arc consistency. Artif. Intell. 159(1–2), 1–26 (2004)
López-Ortiz, A., Quimper, C., Tromp, J., van Beek, P.: A fast and simple algorithm for bounds consistency of the alldifferent constraint. In: Proceedings of IJCAI, Acapulco, Mexico, 9–15 August 2003, pp. 245–250 (2003)
Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic ‘phase transitions’. Nature 400(8), 133–137 (1999)
Montanari, U.: Networks of constraints: fundamental properties and applications to picture processing. Inf. Sci. 7, 95–132 (1974)
Ruan, Y., Kautz, H.A., Horvitz, E.: The backdoor key: a path to understanding problem hardness. In: Proceedings of AAAI, IAAI, San Jose, California, USA, 25–29 July 2004, pp. 124–130 (2004)
Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theoret. Comput. Sci. 47(3), 85–93 (1986). https://doi.org/10.1016/0304-3975(86)90135-0
Williams, R., Gomes, C.P., Selman, B.: Backdoors to typical case complexity. In: Proceedings of IJCAI, Acapulco, Mexico, 9–15 August 2003, pp. 1173–1178 (2003)
Acknowledgements
This research has been funded by Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Escamocher, G., Siala, M., O’Sullivan, B. (2018). From Backdoor Key to Backdoor Completability: Improving a Known Measure of Hardness for the Satisfiable CSP. In: van Hoeve, WJ. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2018. Lecture Notes in Computer Science(), vol 10848. Springer, Cham. https://doi.org/10.1007/978-3-319-93031-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-93031-2_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93030-5
Online ISBN: 978-3-319-93031-2
eBook Packages: Computer ScienceComputer Science (R0)