We study the parameterized complexity of detecting small backdoor sets for instances of the propositional satisfiability problem (SAT). The notion of backdoor sets has been recently introduced by Williams, Gomes, and Selman for explaining the ‘heavy-tailed’ behavior of backtracking algorithms. If a small backdoor set is found, then the instance can be solved efficiently by the propagation and simplification mechanisms of a SAT solver. Empirical studies indicate that structured SAT instances coming from practical applications have small backdoor sets. We study the worst-case complexity of detecting backdoor sets with respect to the simplification and propagation mechanisms of the classic Davis–Logemann–Loveland (DLL) procedure. We show that the detection of backdoor sets of size bounded by a fixed integer k is of high parameterized complexity. In particular, we determine that this detection problem (and some of its variants) is complete for the parameterized complexity class W[P]. We achieve this result by means of a generalization of a reduction due to Abrahamson, Downey, and Fellows.
Key wordssatisfiability unit propagation pure literal elimination backdoor sets parameterized complexity W[P]-completeness
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- 2.Cook, S. A. and Mitchell, D. G. (1997) Finding hard instances of the satisfiability problem: A survey, in Satisfiability problem: theory and applications (Piscataway, NJ, 1996), American Mathematical Society, pp. 1–17.Google Scholar
- 5.Downey, R. G. and Fellows, M. R. (1999) Parameterized Complexity, Monographs in Computer Science. Springer.Google Scholar
- 7.Interian, Y. (2003) Backdoor sets for random 3-SAT, in Sixth International Conference on Theory and Applications of Satisfiability Testing, S. Margherita Ligure, Portofino, Italy, May 5–8, 2003, (SAT 2003), informal proceedings, pp. 231–238.Google Scholar
- 9.Nishimura, N., Ragde, P. and Szeider, S. (2004) Detecting backdoor sets with respect to Horn and binary clauses, in H. Hoos and D. G. Mitchell (eds.), Seventh International Conference on Theory and Applications of Satisfiability Testing, 10–13 May, 2004, Vancouver, BC, Canada (SAT 2004), informal proceedings, pp. 96–103.Google Scholar
- 10.Ruan, Y., Kautz, H. A. and Horvitz, E. (2004) The backdoor key: A path to understanding problem hardness, in D. L. McGuinness and G. Ferguson (eds.), Proceedings of the 19th National Conference on Artificial Intelligence, 16th Conference on Innovative Applications of Artificial Intelligence, pp. 124–130.Google Scholar
- 11.Williams, R., Gomes, C. and Selman, B. (2003a) Backdoors to typical case complexity, in G. Gottlob and T. Walsh (eds.), Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, IJCAI 2003, pp. 1173–1178.Google Scholar
- 12.Williams, R., Gomes, C. and Selman, B. (2003b) On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search, in Sixth International Conference on Theory and Applications of Satisfiability Testing, S. Margherita Ligure, Portofino, Italy, May 5–8, 2003 (SAT 2003), informal proceedings, pp. 222–230.Google Scholar