Exponential Lower Bounds for DPLL Algorithms on Satisfiable Random 3-CNF Formulas

  • Dimitris Achlioptas
  • Ricardo Menchaca-Mendez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7317)


We consider the performance of a number of DPLL algorithms on random 3-CNF formulas with n variables and m = rn clauses. A long series of papers analyzing so-called “myopic” DPLL algorithms has provided a sequence of lower bounds for their satisfiability threshold. Indeed, for each myopic algorithm \({\mathcal A}\) it is known that there exists an algorithm-specific clause-density, \(r_{\mathcal A}\), such that if \(r < r_{\mathcal A}\), the algorithm finds a satisfying assignment in linear time. For example, \(r_{\mathcal A}\) equals 8/3 = 2.66.. for orderred-dll and 3.003... for generalized unit clause. We prove that for densities well within the provably satisfiable regime, every backtracking extension of either of these algorithms takes exponential time. Specifically, all extensions of orderred-dll take exponential time for r > 2.78 and the same is true for generalized unit clause for all r > 3.1. Our results imply exponential lower bounds for many other myopic algorithms for densities similarly close to the corresponding \(r_{\mathcal A}\).


Interpolation Method Satisfying Assignment Partial Assignment Unit Clause Poisson Random Variable 
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  1. 1.
    Achlioptas, D.: Lower bounds for random 3-sat via differential equations. Theoretical Computer Science 265(1-2), 159–185 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Achlioptas, D., Beame, P., Molloy, M.: A sharp threshold in proof complexity yields lower bounds for satisfiability search. Journal of Computer and System Sciences 68(2), 238–268 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Achlioptas, D., Coja-Oghlan, A.: Algorithmic barriers from phase transitions. In: IEEE 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, pp. 793–802. IEEE (2008)Google Scholar
  4. 4.
    Achlioptas, D., Kirousis, L.M., Kranakis, E., Krizanc, D.: Rigorous results for random (2+ p)-SAT. Theoretical Computer Science 265(1), 109–129 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Achlioptas, D., Menchaca-Mendez, R.: Unsatisfiability bounds for random csps from an energetic interpolation method (2012) (to appear in ICALP 2012)Google Scholar
  6. 6.
    Bayati, M., Gamarnik, D., Tetali, P.: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. In: STOC 2010, pp. 105–114 (2010)Google Scholar
  7. 7.
    Beame, P., Pitassi, T.: Propositional proof complexity: Past, present and future. Current Trends in Theoretical Computer Science, 42–70 (2001)Google Scholar
  8. 8.
    Broder, A.Z., Frieze, A.M., Upfal, E.: On the satisfiability and maximum satisfiability of random 3-cnf formulas. In: Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1993, pp. 322–330. Society for Industrial and Applied Mathematics, Philadelphia (1993), Google Scholar
  9. 9.
    Chvátal, V., Reed, B.: Mick gets some (the odds are on his side) [satisfiability]. In: Proceedings 33rd Annual Symposium on Foundations of Computer Science, pp. 620–627. IEEE (1992)Google Scholar
  10. 10.
    Chvatal, V., Szemeredi, E.: Many hard examples for resolution. Journal of the Association for Computing Machinery 35(4), 759–768 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Coja-Oghlan, A.: On belief propagation guided decimation for random k-sat. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 957–966. SIAM (2011)Google Scholar
  12. 12.
    Cook, S., Mitchell, D.: Finding hard instances of the satisfiability problem. In: Satisfiability Problem: Theory and Applications: DIMACS Workshop, March 11-13, vol. 35, p. 1. Amer. Mathematical Society (1997)Google Scholar
  13. 13.
    Díaz, J., Kirousis, L., Mitsche, D., Pérez-Giménez, X.: On the satisfiability threshold of formulas with three literals per clause. Theoretical Computer Science 410(30-32), 2920–2934 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. Journal of Statistical Physics 111(3), 535–564 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Friedgut, E.: Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc. 12, 1017–1054 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Frieze, A., Suen, S.: Analysis of Two Simple Heuristics on a Random Instance ofk-sat. Journal of Algorithms 20(2), 312–355 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Goerdt, A.: A threshold for unsatisfiability. Journal of Computer and System Sciences 53(3), 469–486 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics 230(1), 71–79 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kaporis, A.C., Kirousis, L.M., Lalas, E.G.: The probabilistic analysis of a greedy satisfiability algorithm. Random Structures & Algorithms 28(4), 444–480 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Mézard, M., Mora, T., Zecchina, R.: Clustering of solutions in the random satisfiability problem. Physical Review Letters 94(19), 197205 (2005)CrossRefGoogle Scholar
  21. 21.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297(5582), 812–815 (2002)CrossRefGoogle Scholar
  22. 22.
    Ming-Te, C., Franco, J.: Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the <i> k</i> satisfiability problem. Information Sciences 51(3), 289–314 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Monasson, R., Zecchina, R.: Statistical mechanics of the random k-satisfiability model. Phys. Rev. E 56, 1357–1370 (1997), MathSciNetCrossRefGoogle Scholar
  24. 24.
    Panchenko, D., Talagrand, M.: Bounds for diluted mean-fields spin glass models. Probability Theory and Related Fields 130(3), 319–336 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dimitris Achlioptas
    • 1
    • 2
    • 3
  • Ricardo Menchaca-Mendez
    • 3
  1. 1.University of AthensGreece
  2. 2.CTIGreece
  3. 3.University of CaliforniaSanta CruzUSA

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