# Existence Versus Exploitation: The Opacity of Backdoors and Backbones Under a Weak Assumption

- 1 Citations
- 414 Downloads

## Abstract

Backdoors and backbones of Boolean formulas are hidden structural properties. A natural goal, already in part realized, is that solver algorithms seek to obtain substantially better performance by exploiting these structures.

However, the present paper is not intended to improve the performance of SAT solvers, but rather is a cautionary paper. In particular, the theme of this paper is that there is a potential chasm between the existence of such structures in the Boolean formula and being able to effectively exploit them. This does not mean that these structures are not useful to solvers. It does mean that one must be very careful not to assume that it is computationally easy to go from the existence of a structure to being able to get one’s hands on it and/or being able to exploit the structure.

For example, in this paper we show that, under the assumption that \(\mathrm {P}\ne \mathrm {NP}\), there are easily recognizable families of Boolean formulas with strong backdoors that are easy to find, yet for which it is hard (in fact, NP-complete) to determine whether the formulas are satisfiable. We also show that, also under the assumption \(\mathrm {P}\ne \mathrm {NP}\), there are easily recognizable sets of Boolean formulas for which it is hard (in fact, NP-complete) to determine whether they have a large backbone.

## Notes

### Acknowledgments

We thank the SOFSEM referees for helpful comments. Work done in part while L. Hemaspaandra was visiting ETH-Zürich and U-Düsseldorf.

## References

- 1.Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM
**5**, 394–397 (1962)MathSciNetCrossRefGoogle Scholar - 2.Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM
**7**(3), 201–215 (1960)MathSciNetCrossRefGoogle Scholar - 3.Dilkina, B., Gomes, C., Sabharwal, A.: Tradeoffs in the complexity of backdoors to satisfiability: dynamic sub-solvers and learning during search. Ann. Math. Artif. Intell.
**70**(4), 399–431 (2014)MathSciNetCrossRefGoogle Scholar - 4.Gasarch, W.: The second P =? NP poll. SIGACT News
**43**(2), 53–77 (2012)CrossRefGoogle Scholar - 5.Hemaspaandra, L., Narváez, D.: The opacity of backbones. Technical report, June 2016. arXiv:1606.03634 [cs.AI], Computing Research Repository, arXiv.org/corr/. Accessed June 2017 to December 2018. Revised January 2017
- 6.Hemaspaandra, L., Narváez, D.: The opacity of backbones. In: Proceedings of the 31st AAAI Conference on Artificial Intelligence, pp. 3900–3906. AAAI Press, February 2017Google Scholar
- 7.Hemaspaandra, L., Zimand, M.: Strong self-reducibility precludes strong immunity. Math. Syst. Theory
**29**(5), 535–548 (1996)MathSciNetCrossRefGoogle Scholar - 8.Kilby, P., Slaney, J., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Proceedings of the 20th National Conference on Artificial Intelligence, pp. 1368–1373. AAAI Press (2005)Google Scholar
- 9.Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Informal Proceedings of the 7th International Conference on Theory and Applications of Satisfiability Testing, pp. 96–103, May 2004Google Scholar
- 10.Szeider, S.: Backdoor sets for DLL subsolvers. J. Autom. Reasoning
**35**(1–3), 73–88 (2005)MathSciNetzbMATHGoogle Scholar - 11.Willams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence, pp. 1173–1178. Morgan Kaufmann, August 2003Google Scholar