Approximating the Satisfiability Threshold for Random k -XOR-formulas

TL;DR: In this article, it was shown that the optimization problem MAX 2-SAT undergoes a phase transition in the same way as the decision problem of MAX SAT, and at the same critical value c = 1.

TL;DR: Higher bounds are established on the average distortion achieved by check-regular ensemble of LDGM codes under optimal minimum distance source encoding and a family of message-passing algorithms are proposed, ranging from the standard belief propagation algorithm at one extreme to a variant of survey propagation algorithms at the other.

TL;DR: It is shown that the optimization problem MAX 2-SAT undergoes a phase transition just as the 2- SAT decision problem does, and at the same critical value c = 1, as well as analogous results for random MAX CUT.

TL;DR: An analysis of the average properties of a local search procedure (RandomWalkSAT) for the satisfaction of random Boolean constraints is presented and an annealed calculation of the height zeta(alpha) of this barrier is proposed.

Abstract: Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that a) P is invariant under graph automorphisims. b) If graph H has property P , then so does any graph G having H as a subgraph. A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollobas and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kCNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability. In this paper we indeed introduce a simple condition and prove it is sufficient. Stated roughly, in the setting of random graphs, the main theorem states that if a property has a coarse threshold, then it can be approximated by the property of having certain given graphs as a subgraph. This condition can be applied in a more

TL;DR: The performance of two algorithms, GUC and SC, are considered, when applied to a random instance ?

TL;DR: A new structural (or syntatic) approach for estimating the satisfiability threshold of random 3-SAT formulae is presented, and its efficiency in obtaining a jump from the previous upper bounds is shown.