# Smooth and sharp thresholds for random {k}-XOR-CNF satisfiability

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 37, Issue: 2, page 127-147
- ISSN: 0988-3754

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topCreignou, Nadia, and Daudé, Hervé. "Smooth and sharp thresholds for random {k}-XOR-CNF satisfiability." RAIRO - Theoretical Informatics and Applications 37.2 (2010): 127-147. <http://eudml.org/doc/92718>.

@article{Creignou2010,

abstract = {
The aim of this paper is to study the threshold behavior for the
satisfiability property of a random k-XOR-CNF formula or
equivalently for the consistency of a random Boolean linear
system with k variables per equation. For k ≥ 3 we show the
existence of a sharp threshold for the satisfiability of a random
k-XOR-CNF formula, whereas there are smooth thresholds for k=1
and k=2.
},

author = {Creignou, Nadia, Daudé, Hervé},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Threshold phenomenon; satisfiability; phase transition; random
Boolean linear systems.; threshold phenomenon},

language = {eng},

month = {3},

number = {2},

pages = {127-147},

publisher = {EDP Sciences},

title = {Smooth and sharp thresholds for random \{k\}-XOR-CNF satisfiability},

url = {http://eudml.org/doc/92718},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Creignou, Nadia

AU - Daudé, Hervé

TI - Smooth and sharp thresholds for random {k}-XOR-CNF satisfiability

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 2

SP - 127

EP - 147

AB -
The aim of this paper is to study the threshold behavior for the
satisfiability property of a random k-XOR-CNF formula or
equivalently for the consistency of a random Boolean linear
system with k variables per equation. For k ≥ 3 we show the
existence of a sharp threshold for the satisfiability of a random
k-XOR-CNF formula, whereas there are smooth thresholds for k=1
and k=2.

LA - eng

KW - Threshold phenomenon; satisfiability; phase transition; random
Boolean linear systems.; threshold phenomenon

UR - http://eudml.org/doc/92718

ER -

## References

top- R. Aharoni and N. Linial, Minimal non 2-colorable hypergraphs and minimal unsatisfiable formulas. J. Combin. Theory Ser. A 43 (1986). Zbl0611.05045
- B. Aspvall, M.F. Plass and R.E. Tarjan, A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inform. Process. Lett.8 (1979) 121-123. Zbl0398.68042
- B. Bollobás, Random graphs. Academic Press (1985). Zbl0567.05042
- V. Chvátal, Almost all graphs with 1.44n edges are 3-colorable. Random Struct. Algorithms2 (1991) 11-28. Zbl0715.05026
- V. Chvátal and B. Reed, Mick gets some (the odds are on his side), in Proc. of the 33rd Annual Symposium on Foundations of Computer Science. IEEE (1992) 620-627. Zbl0977.68538
- N. Creignou and H. Daudé, Satisfiability threshold for random XOR-CNF formulæ. Discrete Appl. Math.96-97 (1999) 41-53. Zbl0941.68151
- O. Dubois, Y. Boufkhad and J. Mandler, Typical random 3-SAT formulae and the satisfiability threshold, in Proc. of the 11th ACM-SIAM Symposium on Discrete Algorithms, SODA'2000 (2000) 124-126. Zbl0959.68135
- P. Erdös and A. Rényi, On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci.7 (1960) 17-61. Zbl0103.16301
- E. Friedgut and an Appendix by J. Bourgain, Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc.12 (1999) 1017-1054. Zbl0932.05084
- A. Frieze and S. Suen, Analysis of two simple heuristics on a random instance of k-SAT. J. Algorithms20 (1996) 312-355. Zbl0852.68038
- I.P. Gent and T. Walsh, The SAT phase transition, in Proc. of the 11th European Conference on Artificial Intelligence (1994) 105-109.
- A. Goerdt, A threshold for unsatisfiability. J. Comput. System Sci.53 (1996) 469-486. Zbl0870.68077
- G. Grimmet, Percolation. Springer Verlag (1989).
- S. Janson, Poisson convergence and Poisson processes with applications to random graphs. Stochastic Process. Appl.26 (1987) 1-30. Zbl0633.60070
- S. Kirkpatrick and B. Selman, Critical behavior in the satisfiability of random Boolean expressions. Science264 (1994) 1297-1301. Zbl1226.68097
- V.F. Kolchin, Random graphs and systems of linear equations in finite fields. Random Struct. Algorithms5 (1995) 425-436.
- V.F. Kolchin, Random graphs. Cambridge University Press (1999). Zbl0918.05001
- V.F. Kolchin and V.I. Khokhlov, A threshold effect for systems of random equations of a special form. Discrete Math. Appl.2 (1992) 563-570. Zbl0787.05073
- I.N. Kovalenko, On the limit distribution of the number of solutions of a random system of linear equations in the class of boolean functions. Theory Probab. Appl.12 (1967) 47-56.
- D. Mitchell, B. Selman and H. Levesque, Hard and easy distributions of SAT problems, in Proc. of the 10th National Conference on Artificial Intelligence (1992) 459-465.
- R. Monasson and R. Zecchina, Statistical mechanics of the random K-sat model. Phys. Rev. E56 (1997) 1357. Zbl0942.68052
- T.J. Schaefer, The complexity of satisfiability problems, in Proceedings 10th STOC, San Diego (CA, USA). Association for Computing Machinery (1978) 216-226. Zbl1282.68143
- L. Takács, On the limit distribution of the number of cycles in a random graph. J. Appl. Probab.25 (1988) 359-376.

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