Random k-SAT is the single most intensely studied example of a random constraint satisfaction problem. But despite substantial progress over the past decade, the threshold for the existence of satisfying assignments is not known precisely for any k≥3. The best current results, based on the second moment method, yield upper and lower bounds that differ by an additive k ⋅ ln2 2, a term that is unbounded in k (Achlioptas, Peres: STOC 2003). The basic reason for this gap is the inherent asymmetry of the Boolean values 'true' and 'false' in contrast to the perfect symmetry, e.g., among the various colors in a graph coloring problem. Here we develop a new asymmetric second moment method that allows us to tackle this issue head on for the first time in the theory of random CSPs. This technique enables us to compute the k-SAT threshold up to an additive ln2-1 2+O(1 k) 0.19. Independently of the rigorous work, physicists have developed a sophisticated but non-rigorous technique called the "cavity method" for the study of random CSPs (Mezard, Parisi, Zecchina: Science 2002). Our result matches the best bound that can be obtained from the so-called "replica symmetric" version of the cavity method, and indeed our proof directly harnesses parts of the physics calculations.
For many random constraint satisfaction problems such as random satisfiability or random graph or hypergraph coloring, the best current estimates of the threshold for the existence of solutions are based on the first and the second moment method. However, in most cases these techniques do not yield matching upper and lower bounds. Sophisticated but non-rigorous arguments from statistical mechanics have ascribed this discrepancy to the existence of a phase transition called condensation that occurs shortly before the actual threshold for the existence of solutions and that affects the combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time that a condensation transition exists in a natural random CSP, namely in random hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment method applied to the number of 2-colorings breaks down strictly before the condensation transition. Our proof also yields slightly improved bounds on the threshold for random hypergraph 2-colorability. We consider the problem of @math -colouring a random @math -uniform hypergraph with @math vertices and @math edges, where @math , @math , @math remain constant as @math tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case @math , must have one of two easily computable values as @math tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs. Given d ?? (0,??) let kd be the smallest integer k such that d < 2k log k. We prove that the chromatic number of a random graph G(n, d n) is either kd or kd + 1 almost surely.  We consider a random instance I of k-SAT with n variables and m clauses, where k=k(n) satisfies k—log2 n→∞. Let m 0=2 k nln2 and let ∈=∈(n)>0 be such that ∈n→∞. We prove that @math  Let Fk(n,m) be a random k-SAT formula on n variables formed by selecting uniformly and independently m out of all possible k-clauses. It is well-known that for r ≥ 2k ln 2, Fk(n,rn) is unsatisfiable with probability 1-o(1). We prove that there exists a sequence tk = O(k) such that for r ≥ 2k ln 2 - tk, Fk(n,rn) is satisfiable with probability 1-o(1).Our technique yields an explicit lower bound for every k which for k > 3 improves upon all previously known bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94. Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal @math -SAT is @math . As a corollary, we establish that the threshold for random @math -SAT is of order @math , resolving a long-standing open problem.