Statistical Physics of the Random K-Satis ability Problem

summary by Olivier Dubois and Remi Monasson] The satisfaction of constrained formulae is a key issue in complexity theory. Many computational problems are shown to be NP-complete through a polynomial mapping onto the K-Satissability (SAT) problem. Recently, there has been much interest in a random version of the K-SAT problem deened as follows. Consider N Boolean variables x i , i = 1; : : :; N. Call clause C the logical OR of K randomly chosen variables, each of them being negated or left unchanged with equal probabilities. logical AND of all clauses, F, is said to be satissable if there exists a logical assignment to the x's evaluating F to true, unsatissable otherwise. Numerical experiments have concentrated upon the study of the probability P N (; K) that a given F including M = N clauses be satissable. For large sizes of N, there appears a remarkable behaviour: P seems to reach unity for < c (K) and vanishes for > c (K) 6]. Such an abrupt threshold behaviour, separating a SAT phase from an UNSAT one, has indeed been rigourously connrmed for 2-SAT, which is in P, with c (2) = 1 2, 5]. For larger K 3, K-SAT is in NP and much less is known. The existence of a sharp transition has not been proven yet but precise estimates of the thresholds have been found: c (3) ' 4:25. Moreover, some lower and upper bounds to c (3) (if it exists), l:b: = 3:003 and u:b: = 4:64 respectively have been established 4, 3]. The classical approaches to study the SAT phenomenon threshold are both combinatorial and probabilistic. A statistical physics approach was used in 8, 9]. Such an approach allows properties to be predicted. It has been applied already to random graphs and it has led to large deviation results for the threshold phenomenon of random graphs in addition to previously known results. This approach seems therefore to be powerful. However it proves much harder to apply to the SAT threshold phenomenon. It yields in particular a surprising change concerning the proportion of variables xed in the neighbourhood of the threshold between 2-SAT and 3-SAT. This could partly account for the complexity gap between these two problems. In order to apply the statistical physics approach, the following steps were carried out. First, the energy function corresponding to the K-SAT problem is identiied. The logical …