Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-09T07:01:03.641Z Has data issue: false hasContentIssue false
  • English
  • Français

The Satisfiability Threshold for k-XORSAT

Part of: Theory of computing Probability theory on algebraic and topological structures Combinatorics

Published online by Cambridge University Press:  31 July 2015

BORIS PITTEL  and
GREGORY B. SORKIN
BORIS PITTEL
Affiliation:
Department of Mathematics, Ohio State University, Columbus OH 43210, USA (e-mail: [email protected])
GREGORY B. SORKIN
Affiliation:
Departments of Management and Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider ‘unconstrained’ random k-XORSAT, which is a uniformly random system of m linear non-homogeneous equations in $\mathbb{F}$2 over n variables, each equation containing k ⩾ 3 variables, and also consider a ‘constrained’ model where every variable appears in at least two equations. Dubois and Mandler proved that m/n = 1 is a sharp threshold for satisfiability of constrained 3-XORSAT, and analysed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT.

We show that m/n = 1 remains a sharp threshold for satisfiability of constrained k-XORSAT for every k ⩾ 3, and we use standard results on the 2-core of a random k-uniform hypergraph to extend this result to find the threshold for unconstrained k-XORSAT. For constrained k-XORSAT we narrow the phase transition window, showing that m − n → −∞ implies almost-sure satisfiability, while m − n → +∞ implies almost-sure unsatisfiability.

MSC classification

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 05A16: Asymptotic enumeration
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 236 - 268
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Achlioptas, D. and Molloy, M. (2015) The solution space geometry of random linear equations. Random Struct. Alg. 46 197–231.Google Scholar
[2] Aronson, J., Frieze, A. and Pittel, B. (1998) On maximum matching in sparse random graphs: Karp–Sipser revisited. Random Struct. Alg. 12 111–177.3.0.CO;2-#>CrossRefGoogle Scholar
[3] Balister, P. (2012) Personal communication.Google Scholar
[4] Blömer, J., Karp, R. and Welzl, E. (1997) The rank of sparse random matrices over finite fields. Random Struct. Alg. 10 407–419.Google Scholar
[5] Bollobás, B., Borgs, C., Chayes, J. T., Kim, J.-H. and Wilson, D. B. (2001) The scaling window of the 2-SAT transition. Random Struct. Alg. 18 201–256.CrossRefGoogle Scholar
[6] de Bruijn, N. G. (1981) Asymptotic Methods in Analysis, Dover.Google Scholar
[7] Chvátal, V. and Reed, B. (1992) Mick gets some (the odds are on his side). In 33rd Annual Symposium on Foundations of Computer Science, IEEE Computer Society, pp. 620–627.Google Scholar
[8] Cooper, C. (2000) On the rank of random matrices. Random Struct. Alg. 16 209–232.Google Scholar
[9] Cooper, C. (2004) The cores of random hypergraphs with a given degree sequence. Random Struct. Alg. 25 353–375.Google Scholar
[10] Coppersmith, D., Gamarnik, D., Hajiaghayi, M. T. and Sorkin, G. B. (2004) Random MAX SAT, random MAX CUT, and their phase transitions. Random Struct. Alg. 24 502–545.Google Scholar
[11] Creignon, N. and Daudé, H. (2003) Smooth and sharp thresholds for random k-XOR-CNF satisfiability. Theor. Inform. Appl. 37 127–147.CrossRefGoogle Scholar
[12] Darling, R. W. R., Penrose, M. D., Wade, A. R. and Zabell, S. L. (2014) Rank deficiency in sparse random GF[2] matrices. Electron. J. Probab. 19, 2458 (Paper 83).Google Scholar
[13] Daudé, H. and Ravelomanana, V. (2008) Random 2-XORSAT at the satisfiability threshold. In LATIN 2008: Theoretical Informatics, Vol. 4957 of Lecture Notes in Computer Science, Springer, pp. 12–23.Google Scholar
[14] Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R. and Rink, M. (2010) Tight thresholds for cuckoo hashing via XORSAT. In Proc. 37th International Colloquium on Automata, Languages and Programming: ICALP'10 (Abramsky, S. et al., eds), Springer, pp. 213–225.Google Scholar
[15] Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R. and Rink, M. (2010) Tight thresholds for cuckoo hashing via XORSAT. arXiv:0912.0287v3 Google Scholar
[16] Dubois, O. and Mandler, J. (2002) The 3-XORSAT threshold. In Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science: FOCS 2002, IEEE Computer Society, pp. 769–778.CrossRefGoogle Scholar
[17] Dubois, O. and Mandler, J. (2002) The 3-XORSAT threshold. CR Acad. Sci. Paris, Ser. I 335 963–966.Google Scholar
[18] Fernandez de la Vega, W. (1992) On random 2-SAT. Manuscript.Google Scholar
[19] Friedgut, E. (1999) Necessary and sufficient conditions for sharp thresholds of graph properties, and the k-SAT problem. J. Amer. Math. Soc. 12 1017–1054.Google Scholar
[20] Goerdt, A. (1996) A threshold for unsatisfiability. J. Comput. System Sci. 53 469–486.CrossRefGoogle Scholar
[21] Kim, J. H. (2006) Poisson cloning model for random graphs, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 873–897.Google Scholar
[22] Kolchin, V. F. (1999) Random Graphs , Vol. 53 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.Google Scholar
[23] Mézard, M., Ricci-Tersenghi, F. and Zecchina, R. (2003) Two solutions to diluted p-spin models and XORSAT problems. J. Statist. Phys. 111 505–533.Google Scholar
[24] Mézard, M., Ricci-Tersenghi, F. and Zecchina, R. (2014) Personal communication.Google Scholar
[25] Molloy, M. (2005) Cores in random hypergraphs and Boolean formulas. Random Struct. Alg. 27 124–135.Google Scholar
[26] Pittel, B. (1986) Paths in a random digital tree: Limiting distributions. Adv. Appl. Probab. 18 139–155.Google Scholar
[27] Pittel, B. and Sorkin, G. B. (2012) The satisfiability threshold for k-XORSAT. arXiv:1212.1905v1 Google Scholar
[28] Pittel, B. and Sorkin, G. B. (2012) The satisfiability threshold for k-XORSAT, using an alternative proof. arXiv:1212.3822 Google Scholar
[29] Pittel, B., Spencer, J. and Wormald, N. (1996) Sudden emergence of a giant k-core in a random graph. J. Combin. Theory Ser B 67 111–151.Google Scholar
[30] Pittel, B. and Yeum, J.-A. (2010) How frequently is a system of 2-linear equations solvable? Electron. J. Combin. 17 R92.Google Scholar
[31] Thompson, C. J. (1972) Mathematical Statistical Mechanics, Macmillan.Google Scholar