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Paradigms for Parameterized Enumeration

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Abstract

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First, we define formally different notions of efficient enumeration in the context of parameterized complexity: FPT-enumeration and delayFPT. Second, we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems. Furthermore, we study the parameterized complexity of enumerating all models of Boolean formulas having weight at least k, where k is the parameter, in the famous Schaefer’s framework. We consider propositional formulas that are conjunctions of constraints taken from a fixed finite set Γ. Given such a formula and an integer k, we are interested in enumerating all the models of the formula that have weight at least k. We obtain a dichotomy classification and prove that, according to the properties of the constraint language Γ, either one can enumerate all such models in delayFPT, or no such delayFPT enumeration algorithm exists under some complexity-theoretic assumptions.

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References

  1. Chen, J., Chor, B., Fellows, M.R., Huang, X., Juedes, D.W., Kanji, I.A., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems 201(2), 216–231 (2005)

  2. Creignou, N., Hébrard, J. J.: On generating all solutions of generalized satisfiability problems. Theor. Inf. Appl. 31(6), 499–511 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. Monographs on Discrete Applied Mathematics SIAM Discrete Mathematics and Applications (2001)

  4. Creignou, N., Meier, A., Müller, J. S., Schmidt, J., Vollmer, H.: Paradigms for parameterized enumeration. In: Proceedings International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 8087, pp 290–301. Springer, Berlin Heidelberg (2013)

  5. Creignou, N., Olive, F., Schmidt, J.: Enumerating all solutions of a Boolean CSP by non-decreasing weight. In: Proceedings International Conference on Theory and Applications of Satisfiability Testing, Lecture Notes in Computer Science, vol. 6695, pp 120–133. Springer (2011)

  6. Creignou, N., Vollmer, H.: Boolean constraint satisfaction problems: When does Post’s lattice help?. In: Complexity of Constraints, Lecture Notes in Computer Science, vol. 5250, pp 3–37. Springer (2008)

  7. Damaschke, P.: Parameterized enumeration, transversals, and imperfect phylogeny reconstruction. Theor. Comput. Sci. 351(3), 337–350 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Downey, R.G., Fellows, M.R.: Parameterized Complexity, NY, USA (1999). doi:10.1007/978-1-4612-0515-9

  9. Downey, R.G., Fellows, M.R.: Fundamentals of parameterized complexity. Texts in Computer Science, London, UK (2013). doi:10.1007/978-1-4471-5559-1

    Book  MATH  Google Scholar 

  10. Fernau, H.: On parameterized enumeration. Computing and Combinatorics (2002)

  11. Flum, J., Grohe, M.: Parameterized complexity theory. Springer (2006)

  12. Fomin, F.V., Saurabh, S., Villanger, Y.: A polynomial kernel for proper interval vertex deletion. SIAM J. Discr. Math. 27(4), 1964–1976 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khuller, S., Vazirani, V.V.: Planar graph coloring is not self-reducible, assuming P ≠ NP. Theor. Comput. Sci. 88(1), 183–189 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kratsch, S., Marx, D., Wahlström, M.: Parameterized complexity and kernelizability of max ones and exact ones problems, vol. 6281. Springer (2010)

  16. Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford University Press (2006)

  17. Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Proceedings International Conference on Theory and Applications of Satisfiability Testing, Lecture Notes in Computer Science, vol. 3542 (2004)

  18. Nordh, G., Zanuttini, B.: Frozen boolean partial co-clones. In: Proceedings International Symposium on Multiple-Valued Logic, pp 120–125 (2009)

  19. Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci. 481, 85–99 (2013). doi:10.1016/j.tcs.2012.12.039

    Article  MathSciNet  MATH  Google Scholar 

  20. Samer, M., Szeider, S.: Backdoor trees. In: Proceedings AAAI conference on artificial intelligence, pp 363–368. AAAI Press (2008)

  21. Samer, M., Szeider, S.: Fixed-parameter tractability. In: Biere, A., Heule, M., Van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Chapter 13, pp 425–454 (2009)

  22. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings Annual Symposium on the Theory of Computing, pp 216–226. ACM Press (1978)

  23. Schmidt, J.: Enumeration: Algorithms and complexity. Master’s Thesis, Leibniz Universität Hannover (2009)

  24. Schnorr, C.P.: Optimal algorithms for self-reducible problems. In: Proceedings International Colloquium on Automata, Languages, and Programming, pp 322–337 (1976)

  25. Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) Proceedings of the 6th International Conference Theory and Applications of Satisfiability (SAT’03). Selected and Revised Papers, vol. 2919, pp 188–202 (2004)

  26. Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Proceedings International Joint Conference on Artificial Intelligence, pp 1173–1178 (2003)

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Acknowledgments

We are very thankful to Frédéric Olive for helpful discussions. We also acknowledge many helpful comments from the anonymous reviewers.

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Authors and Affiliations

  1. CNRS, Aix-Marseille Université, LIF UMR 7279, 163 av. de Luminy, 13288, Marseille Cedex 9, France

    Nadia Creignou

  2. Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität Hannover, Appelstr. 4, 30167, Hannover, Germany

    Arne Meier, Julian-Steffen Müller & Heribert Vollmer

  3. Department of Computer and Information Science, Linköping University, SE-581 83, Linköping, Sweden

    Johannes Schmidt

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  1. Nadia Creignou
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  2. Arne Meier
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  3. Julian-Steffen Müller
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  4. Johannes Schmidt
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  5. Heribert Vollmer
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Corresponding author

Correspondence to Arne Meier.

Additional information

A preliminary version of this paper appeared in the proceedings of MFCS 2013, LNCS 8087, pp. 290–301. This work was supported by a Campus France/DAAD Procope grant, Campus France Projet No 28292TE, DAAD Projekt-ID 55892324, and by the French Agence Nationale de la Recherche, AGGREG project reference ANR-14-CE25-0017.

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Creignou, N., Meier, A., Müller, JS. et al. Paradigms for Parameterized Enumeration. Theory Comput Syst 60, 737–758 (2017). https://doi.org/10.1007/s00224-016-9702-4

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  • DOI: https://doi.org/10.1007/s00224-016-9702-4

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