Skip to main content Accessibility help

The Number of Satisfying Assignments of Random Regular k-SAT Formulas



Let Φ be a random k-SAT formula in which every variable occurs precisely d times positively and d times negatively. Assuming that k is sufficiently large and that d is slightly below the critical degree where the formula becomes unsatisfiable with high probability, we determine the limiting distribution of the number of satisfying assignments.



Hide All
[1] Achlioptas, D. and Coja-Oghlan, A. (2008) Algorithmic barriers from phase transitions. In FOCS 2008: 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE, pp. 793802.
[2] Achlioptas, D. and Moore, C. (2006) Random k-SAT: Two moments suffice to cross a sharp threshold. SIAM J. Comput. 36 740762.
[3] Achlioptas, D., Naor, A. and Peres, Y. (2005) Rigorous location of phase transitions in hard optimization problems. Nature 435 759764.
[4] Achlioptas, D. and Peres, Y. (2004) The threshold for random k-SAT is 2k ln 2 - O(k). J. Amer. Math. Soc. 17 947973.
[5] Bapst, V. and Coja-Oghlan, A. (2016) The condensation phase transition in the regular k-SAT model. In RANDOM 2016: 20th International Workshop on Approximation, Randomization, and Combinatorial Optimization, Springer, #22.
[6] Bollobas, B. (2001) Random Graphs, second edition, Cambridge University Press.
[7] Békéssy, A., Békéssy, P. and Komlós, J. (1972) Asymptotic enumeration of regular matrices. Studia Sci. Math. Hungar. 7 343353.
[8] Bolthausen, E. (1984) An estimate of the remainder in a combinatorial central limit theorem. Z. Wahr. Verw. Gebiete 66 379386.
[9] Coja-Oghlan, A. and Panagiotou, K. (2016) The asymptotic k-SAT threshold. Adv. Math. 288 9851068.
[10] Coja-Oghlan, A. and Panagiotou, K. (2013) Going after the k-SAT threshold. In STOC 2013: 45th Annual ACM Symposium on Theory of Computing, ACM, pp. 705–714.
[11] Coja-Oghlan, A. and Vilenchik, D. (2013) Chasing the k-colorability threshold. In FOCS 2013: IEEE 54th Annual Symposium on Foundations of Computer Science, IEEE, pp. 380–389.
[12] Coja-Oghlan, A. and Zdeborová, L. (2012) The condensation transition in random hypergraph 2-coloring. In SODA 2012: 23rd Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 241–250.
[13] Davis, B. and McDonald, D. (1995) An elementary proof of the local central limit theorem. J. Theoret. Probab. 8 693701.
[14] Ding, J., Sly, A. and Sun, N. (2014) Satisfiability threshold for random regular NAE-SAT. In STOC 2014: 46th Annual ACM Symposium on Theory of Computing, ACM, pp. 814–822.
[15] Ding, J., Sly, A. and Sun, N. (2015) Proof of the satisfiability conjecture for large k. In STOC 2015: Proc. 47th Annual ACM Symposium on Theory of Computing, ACM, pp. 59–68.
[16] Frieze, A. and Wormald, N. C. (2005) Random k-Sat: A tight threshold for moderately growing k. Combinatorica 25 297305.
[17] Janson, S. (1995) Random regular graphs: Asymptotic distributions and contiguity. Combin. Probab. Comput. 4 369405.
[18] Kirousis, L., Kranakis, E., Krizanc, D. and Stamatiou, Y. (1998) Approximating the unsatisfiability threshold of random formulas. Random Struct. Alg. 12 253269.
[19] Mézard, M., Parisi, G. and Zecchina, R. (2002) Analytic and algorithmic solution of random satisfiability problems. Science 297 812815.
[20] Molloy, M., Robalewska, H., Robinson, R. W. and Wormald, N. C. (1997) 1-factorisations of random regular graphs. Random Struct. Alg. 10 305321.
[21] Rassmann, F. (2016) On the number of solutions in random hypergraph 2-colouring. Electron. J. Combin. 24 3.11.
[22] Rathi, V., Aurell, E., Rasmussen, L. K. and Skoglund, M. (2010) Bounds on threshold of regular random k-SAT. In SAT 2010: 12th International Conference on Theory and Applications of Satisfiability Testing, Springer, pp. 264–277.
[23] Robinson, R. W. and Wormald, N. C. (1992) Almost all cubic graphs are Hamiltonian. Random Struct. Alg. 3 117125.
[24] Sly, A., Sun, N. and Zhang, Y. (2016) The number of solutions for random regular NAE-SAT. Proc. 57th FOCS 724–731.
[25] Wormald, N. C. (1978) Some problems in the enumeration of labelled graphs. Doctoral thesis, Newcastle University.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed