Glassy Behavior and Jamming of a Random Walk Process for Sequentially Satisfying a Constraint Satisfaction Formula

Random $K$-satisfiability ($K$-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random $K$-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process {\tt SEQSAT}, which satisfies a $K$-SAT formula in a sequential manner. Before satisfying each newly added clause, {\tt SEQSAT} walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density $\alpha$ of the satisfied subformula is less than certain value $\alpha_{cm}$; however it slows down considerably as $\alpha > \alpha_{cm}$ and finally reaches a jammed state at $\alpha \approx \alpha_{j}$. The glassy dynamical behavior of {\tt SEQSAT} for $\alpha \geq \alpha_{cm}$ probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point $\alpha_j$ is larger than the solution space clustering transition point $\alpha_d$, and its value can be predicted by a long-range frustration mean-field theory. For random $K$-SAT with $K\geq 4$, however, our simulation results indicate that $\alpha_j = \alpha_d$. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.Comment: 10 pages, 6 figures, 1 table, a mistake of numerical simulation corrected, and new results adde

Vertex cover problem studied by cavity method: Analytics and population dynamics

We study the vertex cover problem on finite connectivity random graphs by zero-temperature cavity method. The minimum vertex cover corresponds to the ground state(s) of a proposed Ising spin model. When the connectivity c>e=2.718282, there is no state for this system as the reweighting parameter y, which takes a similar role as the inverse temperature \beta in conventional statistical physics, approaches infinity; consequently the ground state energy is obtained at a finite value of y when the free energy function attains its maximum value. The minimum vertex cover size at given c is estimated using population dynamics and compared with known rigorous bounds and numerical results. The backbone size is also calculated.Comment: 7 pages (including 3 figures and 1 table), REVTeX4 forma

Long Range Frustrations in a Spin Glass Model of the Vertex Cover Problem

In a spin glass system on a random graph, some vertices have their spins changing among different configurations of a ground--state domain. Long range frustrations may exist among these unfrozen vertices in the sense that certain combinations of spin values for these vertices may never appear in any configuration of this domain. We present a mean field theory to tackle such long range frustrations and apply it to the NP-hard minimum vertex cover (hard-core gas condensation) problem. Our analytical results on the ground-state energy density and on the fraction of frozen vertices are in good agreement with known numerical and mathematical results.Comment: An erratum is added to the main text. 5 pages, 5 figure

Criticality and Heterogeneity in the Solution Space of Random Constraint Satisfaction Problems

Random constraint satisfaction problems are interesting model systems for spin-glasses and glassy dynamics studies. As the constraint density of such a system reaches certain threshold value, its solution space may split into extremely many clusters. In this paper we argue that this ergodicity-breaking transition is preceded by a homogeneity-breaking transition. For random K-SAT and K-XORSAT, we show that many solution communities start to form in the solution space as the constraint density reaches a critical value alpha_cm, with each community containing a set of solutions that are more similar with each other than with the outsider solutions. At alpha_cm the solution space is in a critical state. The connection of these results to the onset of dynamical heterogeneity in lattice glass models is discussed.Comment: 6 pages, 4 figures, final version as accepted by International Journal of Modern Physics