We identify a new class of hard 3-SAT instances, namely a random 3-SAT problems having exactly one solution and as few clauses as possible. It is numerically shown that the running time of complete methods as well as of local search algorithms for such problems is larger than for random instances around the phase transition point. We therefore provide instances with an exponential complexity in the so-called easy'' region, below the critical value of m n. This puts a new light on the connection between the phase transition phenomenon and NP-completeness.
Most of the studies of random 3-SAT ensemble have been concerned with the computational cost at a constant @math as a function of the ratio @math , where the characteristic phase transition-like curve is observed. This is in a way surprising because for the computational complexity (and also for practical applications) it is the scaling of running time as the problem size increases which is important, i.e. changing @math at fixed @math . Exponential scaling with @math has been numerically observed near the critical point @cite_14 @cite_5 for random 3-SAT as well as above it (albeit with a smaller exponent). Recently the scaling with @math has been studied and the transition from polynomial to exponential complexity has been observed below @math @cite_41 , again for random 3-SAT.
@cite_41 @cite_5 @cite_14
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For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime. A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica symmetric results and also allows for the inclusion of replica symmetry breaking effects. For the 3-SAT problem, we find two transitions as the ratio α of logical clauses per Boolean variables increases. At the first one ( a_s 3.96 ), a non-trivial organization of the solution space in geometrically separated clusters emerges. The multiplicity of these clusters as well as the typical distances between different solutions are calculated. At the second threshold ( a_c 4.48 ), satisfying assignments disappear and a finite fraction ( B_o 0.13 ) of variables are overconstrained and take the same values in all optimal (though unsatisfying) assignments. These values have to be compared to ( a_c 4.27 ), ( B_o 0.4 ) obtained from numerical experiments on small instances. Within the present variational approach, the SAT-UNSAT transition naturally appears as a mixture of a first and a second order transition. For the mixed 2+p-SAT with p<2 5, the behavior is as expected much simpler: a unique smooth transition from SAT to UNSAT takes place at ( a_c = 1 (1 - p) ). 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.