TY - JOUR

T1 - Satisfiability Threshold for Random Regular nae-sat

AU - Ding, Jian

AU - Sly, Allan

AU - Sun, Nike

N1 - Funding Information:
Research supported by J. Ding: NSF Grant DMS-1313596; A. Sly: Sloan Research Fellowship; N. Sun: NDSEG and NSF GRF.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We consider the random regular k-nae- sat problem with n variables, each appearing in exactly d clauses. For all k exceeding an absolute constant k0, we establish explicitly the satisfiability threshold (Formula presented.). We prove that for (Formula presented.) the problem is satisfiable with high probability, while for (Formula presented.) the problem is unsatisfiable with high probability. If the threshold d⋆ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krz̧akała et al. [Proc Natl Acad Sci 104(25):10318–10323, 2007]. Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.

AB - We consider the random regular k-nae- sat problem with n variables, each appearing in exactly d clauses. For all k exceeding an absolute constant k0, we establish explicitly the satisfiability threshold (Formula presented.). We prove that for (Formula presented.) the problem is satisfiable with high probability, while for (Formula presented.) the problem is unsatisfiable with high probability. If the threshold d⋆ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krz̧akała et al. [Proc Natl Acad Sci 104(25):10318–10323, 2007]. Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.

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U2 - 10.1007/s00220-015-2492-8

DO - 10.1007/s00220-015-2492-8

M3 - Article

AN - SCOPUS:84953351828

SN - 0010-3616

VL - 341

SP - 435

EP - 489

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 2

ER -