Phase Transition for Maximum Not-All-Equal Satisfiability

Phase transition is a dramatic transition from one state to another state when a particular parameter varies. This paper aims to study the phase transition of maximum not-all-equal satisfiability problem (Max NAE SAT), an optimization of not-all-equal satisfiability problem (NAE SAT). Given a conjunctive normal formula (CNF) F with n variables and rn k-clauses (the clause exactly contains k literals), we use first-moment method to obtain an upper bound for f(n, rn) the expectation of the maximum number of NAE-satisfied clauses of random Max NAE k-SAT. In addition, we also consider the phase transition of decision version of random Max NAE k-SAT—bounded not-all-equal satisfiability problem (NAE k-SAT(b)). We demonstrate that there is a phase transition point \(r_{k,b}\) separating the region where almost all NAE k-SAT(b) instances can be solved from the region where almost all NAE k-SAT(b) instances can’t be solved. Furthermore, we analyze the upper bound and lower bound for \(r_{k,b}\).

[1]  Simon de Givry,et al.  A logical approach to efficient Max-SAT solving , 2006, Artif. Intell..

[2]  Ping Huang,et al.  An upper (lower) bound for Max (Min) CSP , 2013, Science China Information Sciences.

[3]  Wei Li,et al.  The SAT phase transition , 1999, ArXiv.

[4]  David Gamarnik,et al.  Random MAX SAT, random MAX CUT, and their phase transitions , 2003 .

[5]  Lei Zhang,et al.  Robust low-rank tensor factorization by cyclic weighted median , 2014, Science China Information Sciences.

[6]  Yun Fan,et al.  A general model and thresholds for random constraint satisfaction problems , 2012, Artif. Intell..

[7]  Wei-Tek Tsai,et al.  Software-as-a-service (SaaS): perspectives and challenges , 2013, Science China Information Sciences.

[8]  Weixiong Zhang,et al.  Phase Transitions and Backbones of 3-SAT and Maximum 3-SAT , 2001, CP.

[9]  Toby Walsh,et al.  The TSP Phase Transition , 1996, Artif. Intell..

[10]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[11]  Phokion G. Kolaitis,et al.  Phase Transitions of Bounded Satisfiability Problems , 2003, IJCAI.

[12]  Ke Xu,et al.  A tighter upper bound for random MAX 2-SAT , 2011, Inf. Process. Lett..

[13]  Allan Sly,et al.  The number of solutions for random regular NAE-SAT , 2016, Probability Theory and Related Fields.

[14]  Konstantinos Panagiotou,et al.  Catching the k-NAESAT threshold , 2011, STOC '12.

[15]  Cristopher Moore,et al.  The phase transition in 1-in-k SAT and NAE 3-SAT , 2001, SODA '01.

[16]  Wei Li,et al.  Exact Phase Transitions in Random Constraint Satisfaction Problems , 2000, J. Artif. Intell. Res..

[17]  Rémi Monasson,et al.  Determining computational complexity from characteristic ‘phase transitions’ , 1999, Nature.

[18]  Xiangtao Li,et al.  Phase Transitions of EXPSPACE-Complete Problems: a Further Step , 2012, Int. J. Found. Comput. Sci..

[19]  Assaf Naor,et al.  Rigorous location of phase transitions in hard optimization problems , 2005, Nature.

[20]  Minghao Yin,et al.  Experimental analyses on phase transitions in compiling satisfiability problems , 2014, Science China Information Sciences.

[21]  Cristopher Moore,et al.  The asymptotic order of the random k-SAT threshold , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[22]  Allan Sly,et al.  Satisfiability Threshold for Random Regular nae-sat , 2013, Communications in Mathematical Physics.