Catching the k-NAESAT Threshold

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 (Mezard, 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 2^{k-1}\ln2-(\frac{\ln2}2+\frac14)+\eps_k, where |\eps_k| \le 2^{-(1-o_k(1))k}, thereby verifying the statistical mechanics conjecture for this problem

Going after the k-SAT Threshold

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 $k\geq3$. The best current results, based on the second moment method, yield upper and lower bounds that differ by an additive $k\cdot \frac{\ln2}2$, a term that is unbounded in $k$ (Achlioptas, Peres: STOC 2003). The basic reason for this gap is the inherent asymmetry of the Boolean value `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 $\ln2-\frac12+O(1/k)\approx 0.19$. Independently of the rigorous work, physicists have developed a sophisticated but non-rigorous technique called the "cavity method" for the study of random CSPs (M\'ezard, 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

Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure