Abstract
The satisfiability problem and algorithms for solving it have received greatly increased attention in the last few years. This interest comes from a variety of disciplines such as Computer Science, Operations Research, Graph Theory, and Physics, among others. This paper shows some of the current interesting directions in satisfiability research from these disciplines and presents some possible future directions.
Similar content being viewed by others
References
D. Achlioptas, Setting 2 variables at a time yields a new lower bound for random 3-sat, Manuscript (1999).
R. Aharoni and N. Linial, Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas, Journal of Combinatorial Theory. Series A 43 (1986) 196-204.
B. Aspvall, M.F. Plass and R.E. Tarjan, A linear-time algorithm for testing the truth of certain quantified Boolean formulas, Information Processing Letters 8 (1979) 121-132.
P. Beame, R. Karp, T. Pitassi and M. Saks, On the complexity of unsatisfiability proofs for random k-CNF formulas, in: Proc. of ACM Symposium on the Theory of Computation (1998).
P. Beame and T. Pitassi, Simplified and improved resolution lower bounds, in: Proc. of 37th Symposium on Foundations of Computer Science (1996) pp. 274-282.
E. Ben-Sasson and A. Widgerson, Short proofs are narrow: resolution made simple, Technical Report TR99-022, ECCC Electronic Colloqium on Computational Complexity (1999).
B. Bollobas, C. Borgs, J. Chayes, J.H. Kim and D.B. Wilson, The scaling window of the 2-SAT transition, Random Structures Algorithms, to appear.
E. Boros, Y. Crama and P.L. Hammer, Polynomial-time inference of all valid implications for Horn and related formulae, Annals of Mathematics and Artificial Intelligence 1 (1990) 21-32.
E. Boros, Y. Crama, P.L. Hammer and M. Saks, A complexity index for satisfiability problems, SIAM Journal on Computing 23 (1994) 45-49.
E. Boros, P.L. Hammer and X. Sun, Recognition of q-Horn formulae in linear time, Discrete Applied Mathematics 55 (1994) 1-13.
A.Z. Broder, A.M. Frieze and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, in: Proc. of 4th Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, TX, 1993 (ACM, New York, 1993) pp. 322-330.
C.A. Brown and P.W. Purdom, An average time analysis of backtracking, SIAM Journal on Computing 10 (1981) 583-593.
H.K. Buening, On generalized Horn formulas and k-resolution, Theoretical Computer Science 116 (1993) 405-413.
H.K. Buening, On the minimal unsatisfiability problem for some subclasses of CNF, in: Abstracts of 16th International Symposium on Mathematical Programming, Lausanne (1997).
K.M. Bugrara, Y.F. Pan and P. Purdom, Exponential average time for the pure literal rule, SIAM Journal on Computing 18 (1989) 409-418.
M. Buro and H.K. Buening, Report on a SAT competition, Technical Report FB-17, Mathematik/ Informatik, Universitat Paderborn (November 1992).
V. Chandru and J.N. Hooker, Extended Horn sets in propositional logic, Journal of the Association for Computing Machinery 38 (1991) 205-221.
M.-T. Chao and J. Franco, Probabilistic analysis of two heuristics for the 3-satisfiability problem, SIAM Journal on Computing 15 (1986) 1106-1118.
M.-T. Chao and J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisfiability problem, Information Sciences 51 (1990) 289-314.
V. Chvatal and B. Reed, Mick gets some (the odds are on his side), in: Proc. of 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, PA, 1992 (IEEE Computer Society Press, Los Alamitos, CA, 1992) pp. 620-627.
V. Chvatal and E. Szemeredi, Many hard examples for resolution, Journal of the Association for Computing Machinery 35 (1988) 759-768.
M. Conforti, G. Cornuejols, A. Kapoor, K. Vuskovic and M.R. Rao, Balanced matrices, in: Mathematical Programming: State of the Art, eds. J.R. Birge and K.G. Murty, Braun-Brumfield, USA. Produced in association with the 15th International Symposium on Mathematical Programming, University of Michigan, 1994.
M. Dalal and D.W. Etherington, A hierarchy of tractible satisfiability problems, Information Processing Letters 44 (1992) 173-180.
M. Davis, G. Logemann and D. Loveland, A machine program for theorem-proving, Communications of the ACM 5 (1962) 394-397.
M. Davis and H. Putnam, A computing procedure for quantification theory, Journal of the Association for Computing Machinery 7 (1960) 201-215.
W.F. De la Vega, On random 2-sat, Manuscript (1992).
W.F. Dowling and J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulae, Journal of Logic Programming 1 (1984) 267-284.
R.G. Downey and M.R. Fellows, Fixed parameter tractability and NP-completeness, Congressus Numeratus 87 (1992) 161-178.
O. Dubois and Y. Boufkhad, A general upper bound for the satisfiability threshold of random r-SAT formulae, Journal of Algorithms 25 (1997) 395-420.
O. Dubois, Y. Boufkhad and J. Mandler, Typical random 3-SAT formulae and the satisfiability threshold, in: Proc. of ACM-SIAM Symposium on Discrete Algorithms (2000).
J. Franco, On the probabilistic performance of algorithms for the satisfiability problem, Information Processing Letters 23 (1986) 103-106.
J. Franco, Elimination of infrequent variables improves average case performance of satisfiability algorithms, SIAM Journal on Computing 20 (1991) 1119-1127.
J. Franco, J. Goldsmith, J. Schlipf, E. Speckenmeyer and R.P. Swaminathan, An algorithm for the class of pure implicational formulas, Discrete Applied Mathematics 96-97 (1999) 89-106.
J. Franco and Y.C. Ho, Probabilistic performance of a heuristic for the satisfiability problem, Discrete Applied Mathematics 22 (1988/1989) 35-51.
J. Franco and M. Paull, Probabilistic analysis of the Davis-Putnam procedure for solving the satisfiability problem, Discrete Applied Mathematics 5 (1983) 77-87.
E. Friedgut, Necessary and sufficient conditions for sharp thresholds of graph properties, and the k-SAT problem, Journal of the American Mathematics Society 12 (1999) 1017-1054.
A.M. Frieze and S. Suen, Analysis of two simple heuristics on a random instance of k-SAT, Journal of Algorithms 20 (1996) 312-355.
G. Gallo and M.G. Scutella, Polynomially solvable satisfiability problems, Information Processing Letters 29 (1988) 221-227.
A.V. Gelder, A satisfiability tester for non-clausal propositional calculus, Information and Computation 79 (1988) 1-21.
A. Goerdt, A threshold for unsatisfiability, Journal of Computer System Science 53 (1996) 469-486.
A. Goldberg, P.W. Purdom and C.A. Brown, Average time analysis of simplified Davis-Putnam procedures, Information Processing Letters 15 (1982) 72-75. (Some printer errors in this paper were corrected in Information Processing Letters 16 (1983) 213.)
A. Haken, The intractability of resolution, Theoretical Computer Science 39 (1985) 297-308.
P. Heusch, The complexity of the falsifiability problem for pure implicational formulas, in: Proc. of 20th International Symposium on Mathematical Foundations of Computer Science (MFCS '94), Prague, Czech Republic, eds. J. Wiedermann and P. Hajek, Lecture Notes in Computer Science, Vol. 969 (Springer-Verlag, Berlin, 1995) pp. 221-226.
A. Itai and J. Makowsky, On the complexity of Herbrand's theorem, Working paper 243, Department of Computer Science, Israel Institute of Technology (1982).
L.M. Kirousis, E. Kranakis, D. Krizanc and Y. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures Algorithms 12 (1998) 253-269.
O. Kullman, A first summary on minimal unsatisfiable clause-sets, Technical Report, University of Toronto (June 1998).
O. Kullmann, New methods for 3-SAT decision and worst-case analysis, Theoretical Computer Science 223 (1999) 1-72.
O. Kullman, Investigating a general hierarchy of polynomially decidable classes of CNF's based on short tree-like resolution proofs, Technical Report, University of Toronto (August 1999).
R. Monasson, Statistical mechanics analysis of phase transition and average complexity in random SAT problems, in: Proc of 3rd Workshop on Satisfiability, Renesse, The Netherlands (2000).
B. Monien and E. Speckenmeyer, Solving satisfiability in less than 2n steps, Discrete Applied Mathematics 10 (1983) 117-133.
D. Pretolani, Hierarchies of polynomially solvable satisfiability problems, Annals of Mathematics and Artificial Intelligence 17 (1996) 339-357.
P.W. Purdom and G.N. Haven, Probe order backtracking, SIAM Journal on Computing 26 (1997) 456-483.
I. Schiermeyer, Solving 3-Satisfiability in less than O(1:579n) steps, Lecture Notes in Computer Science, Vol. 702 (1993) pp. 379-394.
I. Schiermeyer, Pure literal lookahead: an O(1:497n) 3-Satisfiability algorithm, in: Proc. of the Workshop on Satisfiability, Universitat delgi Studi, Siena, Italy (1996) 63-72.
J.S. Schlipf, F. Annexstein, J. Franco and R. Swaminathan, On finding solutions for extended Horn formulas, Information Processing Letters 54 (1995) 133-137.
M.G. Scutella, A note on Dowling and Gallier's top-down algorithm for propositional Horn satis-fiability, Journal of Logic Programming 8 (1990) 265-273.
B. Selman, D.G. Mitchell and H.J. Levesque, Generating hard satisfiability problems, Artificial Intelligence 81 (1996) 17-29.
C.A. Tovey, A simplified NP-complete satisfiability problem, Discrete Applied Mathematics 8 (1984) 85-89.
K. Truemper, Monotone decomposition of matrices, Technical Report UTDCS-1-94, University of Texas at Dallas (1994).
K. Truemper, Effective Logic Computation (Wiley, New York, 1998).
A. Urquhart, Hard examples for resolution, Journal of the Association for Computing Machinery 34 (1987) 209-219.
H. van Maaren, Elliptical approximations of propositional formulae, Discrete Applied Mathematics, to appear; Currently available as Technical Report 96-95, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands (1996).
B.W. Wah and Z. Wu, Efficient constrained formulations and trap-avoidance strategies for solving hard satisfiability problems, in: Proc. of 3rd Workshop on Satisfiability, Renesse, The Netherlands (2000).
J.P. Warners, Nonlinear approaches to Satisfiability problems, Ph.D. Thesis, Technical University Eindhoven, The Netherlands (1999).
Rights and permissions
About this article
Cite this article
Franco, J. Some interesting research directions in satisfiability. Annals of Mathematics and Artificial Intelligence 28, 7–15 (2000). https://doi.org/10.1023/A:1018983601518
Issue Date:
DOI: https://doi.org/10.1023/A:1018983601518