Merge pull request #40 from kirousis51/main

 The complexity threshold of random 3-SAT

Author: teorth

constants/52a.md

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+# The complexity threshold of random 3-SAT
+
+## Description of constant
+
+Let $m,n$ be positive integers and let $V$ be a set of $n$ Boolean variables. By a random formula of density $r= m/n$, we mean a collection of $m$ clauses selected u.a.r. with replacement from the set of $8{{n}\choose{3}}$ clauses on three distinct variables from $V$. For linguistic convenience, formulas with variables from $V$ will be called $n$-formulas.  
+
+It is conjectured, and corroborated by experimental results (see e.g. [LT1992]) and non-rigorous considerations of Statistical Physics (see e.g. [MZ1997]), that there is a constant $r_3 \approx 4.2$, here to be denoted also by $C_{52}$, such that for any constant $r$:  
+$$\mbox{if } r > r_3, \mbox{ then } \lim_{n \rightarrow \infty}\Pr[\mbox{an } n\mbox{-formula with density } r  \mbox{ is satisfiable}] =0,$$ 
+whereas
+$$\mbox{if } r < r_3, \mbox{ then } \lim_{n \rightarrow \infty}\Pr[\mbox{an } n\mbox{-formula with density } r \mbox{ is satisfiable}] =1.$$
+
+It has been proved by Friedgut [F1999] that there is a sequence $r_{3,n}, n=1, \dots$ such that for any $\epsilon>0$
+$$\lim_{n \rightarrow \infty}\Pr[
+\mbox{an } n\mbox{-formula with density } \geq (r_{3,n} + \epsilon)  \mbox{ is satisfiable}] =0,$$
+whereas 
+$$\lim_{n \rightarrow \infty}\Pr[
+\mbox{an } n\mbox{-formula with density } \leq (r_{3,n} -\epsilon)  \mbox{ is satisfiable}] =1.$$
+
+Below we give the rigorously proved  upper bounds for $\limsup_{n \rightarrow \infty} r_{3,n}$ and the rigorously proved   lower bounds for 
+$\liminf_{n \rightarrow \infty} r_{3,n}.$
+
+## Known upper bounds
+
+| Bound | Reference  | Comments |
+|---    |---         |---       |
+| 5.191 | [FP1983]   | Direct first moment method |
+| 5.081 | [MV1995]   ||
+| 4.758 | [KMPS1995] ||
+| 4.643 | [DB1997]   ||
+| 4.506 | [DBM2000]  ||
+| 4.596 | [JSV2000]  ||
+| 4.571 | [KKSVZ2007]||
+| 4.453 | [MS2008]   |Under an extra hypothesis,<br> see Additional Comments (1) below  |
+| 4.490 | [DKMP2009] ||
+
+
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+|2.9   | [CF1986].  | See Additional Comments (2) below|
+|2/3 | [CR1992] ||
+|1.63| [BFU1993] ||
+|3.003| [FS1996]||
+|3.145|[A2000]||
+|3.26|[AS2000]||
+|3.42 | [KKL2002]||
+| 3.52| [HS2003], [KKL2003]||
+
+
+## Additional comments
+1. The extra hypothesis used in [MS2008] is about the satisfying assignments of formulas with density below and close to the threshold. This hypothesis has been proved  for $k$-SAT, $k \geq 8.$
+
+2. In [CF1986], the probability of satisfiability is shown to be only a positive constant. However, this, by Friedgut's result of 1999, implies that the probability is actually 1.
+
+
+## References
+- [LT1992] T. Larrabee and Y. Tsuji, Evidence for satisfiability threshold for random 3CNF
+formulas, Tech. Rep. UCSC-CRL-92-42, University of California, Santa Cruz, 1992.
+
+- [MZ1997] R. Monasson and R. Zecchina,  Statistical mechanics of the random $K$-satisfiability model, *Physical Review E* 56(2), 1357-1370, 1997.
+
+- [F1999] E. Friedgut, appendix by J. Bourgain, Sharp thresholds of graph properties, and the $k$-SAT problem, *J. Amer. Math. Soc.* 12,  1017-1054, 1999. 
+
+- [FP1983] J. Franco and M. Paull, Probabilistic analysis of the Davis Putman procedure for
+solving the satisfiability problem, *Discrete Appl. Math.* 5,  77-87, 1983. 
+
+- [MV1995] A. El Maftouhi, and W.F. De La Vega, On random 3-SAT, *Combinatorics, Probability and Computing* 4(3), 189-195, 1995.
+
+- [KMPS1995] A. Kamath, R. Motwani, K. Palem, and P. Spirakis, Tail bounds for occupancy and the
+satisfiability threshold conjecture,  *Random Structures & Algorithms* 7(1), 59-80, 1995.
+
+- [DB1997] O. Dubois, Y. Boufkhad, A general upper bound for the satisfiability threshold of random $r$-SAT formulae, *Journal of Algorithms* 24(2), 395-420, 1997.
+
+- [DBM2000] O. Dubois, Y Boufkhad, and J. Mandler, Typical random 3-SAT formulae and the satisfiability threshold, *Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, 2000*. Also in arXiv preprint: cs/0211036, 2002.
+
+- [JSV2000] S. Janson, Y.C. Stamatiou, M. Vamvakari, Bounding the unsatisfiability threshold of random 3-SAT, *Random Structures & Algorithms* 17(2), 103-116, 2000. 
+
+- [KKSVZ2007] A. Kaporis, L.M. Kirousis, Y.C. Stamatiou, M. Vamvakari, and M. Zito. The unsatisfiability threshold revisited, *Discrete Appl. Math.* 155(12), 1525-1538, 2007.
+
+- [MS2008] E. Maneva, and A. Sinclair,  On the satisfiability threshold and clustering of solutions of random 3-SAT formulas, *Theoretical Computer Science* 407(1-3), 359-369, 2008.
+
+- [DKMP2009] J. Díaz, L. Kirousis, D. Mitsche, and X. Pérez-Giménez, On the satisfiability threshold of formulas with three literals per clause,  *Theoretical Computer Science* 410(30-32), 2920-2934, 2009.
+
+- [CF1986] M-T. Chao, and J. Franco, Probabilistic analysis of two heuristics for the 3-satisfiability problem, *SIAM Journal on Computing* 15(4), 1106-1118, 1986.
+
+- [CR1992] V. Chvátal, and B. Reed, Mick gets some (the odds are on his side),  *Proceedings, 33rd Annual Symposium on Foundations of Computer Science*, IEEE Computer Society, 620-627, 1992.
+
+- [BFU1993] A.Z. Broder, A.M. Frieze, and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas. In *SODA '93*, 322-330, 1993.
+
+- [FS1996] A. Frieze, and S. Suen, Analysis of two simple heuristics on a random instance of $k$-SAT, *Journal of Algorithms* 20(2), 312-355, 1996.
+
+- [A2000] D. Achlioptas, Setting 2 variables at a time yields a new lower bound for random 3-SAT. *Proceedings of the thirty-second annual ACM symposium on Theory of computing*, 28-37,  2000. 
+
+- [AC2000] D. Achioptas, and G.B. Sorkin, Optimal myopic algorithms for random 3-SAT, *Proceedings 41st Annual Symposium on Foundations of Computer Science*, IEEE Computer Society, 590-600, 2000.
+
+- [KKL2000] A.C. Kaporis, L.M. Kirousis, and E.G. Lalas, The probabilistic analysis of a greedy satisfiability algorithm, *Algorithms - ESA*, 574-586, 2002. 
+
+- [HS2003] M. Hajiaghayi, and G.B. Sorkin, The satisfiability threshold of random 3-SAT is at least 3.52, arXiv preprint math/0310193, 2003.
+
+- [KKL2003] A.C. Kaporis, L.M. Kirousis, and E.G. Lalas, Selecting complementary pairs of literals,  *Electronic Notes in Discrete Mathematics* 16, 47-70,  2003.