The best current estimates of the thresholds for the existence of solutions in random constraint satisfaction problems ('CSPs') mostly derive from the first and the second moment method. Yet apart from a very few exceptional cases these methods do not quite yield matching upper and lower bounds. According to deep but non-rigorous arguments from statistical mechanics, this discrepancy is due to a change in the geometry of the set of solutions called condensation that occurs shortly before the actual threshold for the existence of solutions (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). To cope with condensation, physicists have developed a sophisticated but non-rigorous formalism called Survey Propagation (Me-zard, Parisi, Zecchina: Science 2002). This formalism yields precise conjectures on the threshold values of many random CSPs. Here we develop a new Survey Propagation inspired second moment method for the random k-NAESAT problem, which is one of the standard benchmark problems in the theory of random CSPs. This new technique allows us to overcome the barrier posed by condensation rigorously. We prove that the threshold for the existence of solutions in random k-NAESAT is 2k-1ln2-(ln 2 2+1 4)+ek, where |ek| ≤ 2-(1-ok(1))k, thereby verifying the statistical mechanics conjecture for this problem.
@cite_18: 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.
@cite_10: 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.
@cite_8: A 2-coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. Let H = H(k, n, p) be a random k-uniform hypergraph on a vertex set V of cardinality n, where each k-subset of V is an edge of H with probability p, independently of all other k-subsets. Let m = p(????) denote the expected number of edges in H. Let us say that a sequence of events En holds with high probability (w.h.p.) if limn → ∞ Pr[En] = 1. It is easy to show that if m = c2kn then w.h.p H is not 2-colorable for c > ln 2 2. We prove that there exists a constant c > 0 such that if m = (c2k k)n, then w.h.p H is 2-colorable.
The proofs of the lower bounds in @cite_10 @cite_18 and in the present paper are non-constructive in the sense that they do not entail an efficient algorithm for finding a NAE-solution The best current algorithm for random @math -NAESAT is known to succeed for @math , a factor of @math below @math @cite_8 .