During the past decade, phase-transition phenomena in the random 3-satisfiability (3-SAT) problem has been intensively studied by statistical physics methods. In this work, we study the random 3-SAT problem by the mean-field first-step replica-symmetry-broken cavity theory at the limit of temperature $T\to 0$. The reweighting parameter $y$ of the cavity theory is allowed to approach infinity together with the inverse temperature $β$ with fixed ratio $r=y / β$. Focusing on the the system's space of satisfiable configurations, we carry out extensive population dynamics simulations using the technique of importance sampling and we obtain the entropy density $s(r)$ and complexity $Σ(r)$ of zero-energy clusters at different $r$ values. We demonstrate that the population dynamics may reach different fixed points with different types of initial conditions. By knowing the trends of $s(r)$ and $Σ(r)$ with $r$, we can judge whether a certain type of initial condition is appropriate at a given $r$ value. This work complements and confirms the results of several other very recent theoretical studies.
Random $K$-satisfiability ($K$-SAT) is a paradigmatic model system for studying phase transitions in constraint satisfaction problems and for developing empirical algorithms. The statistical properties of the random $K$-SAT solution space have been extensively investigated, but most earlier efforts focused on solutions that are typical. Here we consider maximally flexible solutions which satisfy all the constraints only using the minimum number of variables. Such atypical solutions have high internal entropy because they contain a maximum number of null variables which are completely free to c…
Exponentially hard problems are sometimes polynomial, a large deviation analysis of search algorithms for the random Satisfiability problem, and its application to stop-and-restart resolutions
Asymmetry in the synaptic interactions between neurons plays a crucial role in determining the memory storage and retrieval properties of recurrent neural networks. In this work, we analyze the problem of storing random memories in a network of neurons connected by a synaptic matrix with a definite degree of asymmetry. We study the corresponding satisfiability and clustering transitions in the space of solutions of the constraint satisfaction problem associated with finding synaptic matrices given the memories. We find, besides the usual SAT/UNSAT transition at a critical number of memories to…
We study local search algorithms to solve instances of the random $k$-satisfiabi lity problem, equivalent to finding (if they exist) zero-energy ground states of statistical models with disorder on random hypergraphs. It is well known that the best such algorithms are akin to non-equilibrium processes in a high-dimensional space. In particular, algorithms known as focused, and which do not obey detailed balance, outperform simulated annealing and related methods in the task of finding the solution to a complex satisfiability problem, that is to find (exactly or approximately) the minimum in a …
Planting a solution into the random RB model, which is a prototype of random constraint satisfaction problem (CSP) with growing domains, can generate very hard satisfiable CSP benchmarks. We study the solution space structure of the planted RB model. With constraint density growing, we find that this model goes through four phase transitions. In the replica symmetric phase, what we call the independent phase transition occurs, after which the planted cluster (cluster containing the planted solution) is separated from the giant cluster. Then the solutions except that in the planted cluster go t…
Here we study the NP-complete $K$-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random $K$-SAT, with $α=M/N $ as control parameter, can be solved quickly for small enough values of $α$. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically…
For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satisfiability transition occurs. For random $k$-SAT and related models it happens at clause density $α$ around $2^k$. Just below the threshold, further results suggest that the solution space has a "1RSB" structure of a large bounded number of near-orthogonal clusters inside the space of variable assignments $\{0,1\}^N$. In the unsatisfiable regime, it is natural to consider max-satisfiability: violating the least number of constraints. For a simplified variant, the st…
A detailed Monte Carlo-study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies $α_3 \leq 4.262$. If the assumption is correct, this means that the actual threshold value for $k=3$ is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large $k$. Our result thus indicate that there are distinct behaviors for $k$ above and below some critical $k_c$, and the cavity method may provide a correct mean-field pictur…
