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.
Other problems where the second moment method succeeds are symmetric as well. Pioneering the use of the second moment method in random CSPs, Achlioptas and Moore @cite_11 computed the random @math -NAESAT threshold within an additive @math . By enhancing this argument with insights from physics this gap can be narrowed to a mere @math @cite_19 @cite_18 . Moreover, the best current bounds on the random (hyper)graph @math -colorability thresholds are based on vanilla'' second moment arguments as well @cite_29 @cite_26 . In summary, in all the previous second moment arguments, the issue of asymmetry either did not appear at all by the nature of the problem @cite_11 @cite_29 @cite_19 @cite_18 @cite_16 @cite_26 @cite_15 , or it was sidestepped @cite_25 .
@cite_18 @cite_26 @cite_29 @cite_19 @cite_15 @cite_16 @cite_25 @cite_11
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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. Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random @math -SAT or @math -NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph @math -coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of @math -colorings, allows us to establish a substantially improved lower bound on the @math -colorability threshold. The new lower bound is within an additive @math of a simple first-moment upper bound and within @math of the physics conjecture. By comparison, the best previous lower bound left a gap of about @math , unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. 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. In this paper we establish a substantially improved lower bound on the k-color ability threshold of the random graph G(n, m) with n vertices and m edges. The new lower bound is ≈ 1.39 less than the 2k ln (k)-ln (k) first-moment upper bound (and approximately 0.39 less than the 2k ln (k) - ln(k) - 1 physics conjecture). By comparison, the best previous bounds left a gap of about 2+ln(k), unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments [: PNAS 2007]. Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of k-colorings of the random graph.