2016/02/16 by Róbert Sumi, Melinda Varga, Sumi, Róbert +5 · 1 citation
Chemistry · #Mass Spectrometry Techniques and Applications
paper · pdf · doi:10.48550/arxiv.1602.05152
Transient chaos is an ubiquitous phenomenon characterizing the dynamics of\nphase space trajectories evolving towards a steady state attractor in physical\nsystems as diverse as fluids, chemical reactions and condensed matter systems.\nHere we show that transient chaos also appears in the dynamics of certain\nefficient algorithms searching for solutions of constraint satisfaction\nproblems that include scheduling, circuit design, routing, database problems or\neven Sudoku. In particular, we present a study of the emergence of hardness in\nBoolean satisfiability (k-SAT), a canonical class of constraint satisfaction\nproblems, by using an analog deterministic algorithm based on a system of\nordinary differential equations. Problem hardness is defined through the escape\nrate \κ, an invariant measure of transient chaos of the dynamical system\ncorresponding to the analog algorithm, and it expresses the rate at which the\ntrajectory approaches a solution.We show that for a given density of\nconstraints and fixed number of Boolean variables N, the hardness of formulas\nin random k-SAT ensembles has a wide variation, approximable by a lognormal\ndistribution. We also show that when increasing the density of constraints\n\α, hardness appears through a second-order phase transition at\n\α\χ in the random 3-SAT ensemble where dynamical trajectories\nbecome transiently chaotic. A similar behavior is found in 4-SAT as well,\nhowever, such transition does not occur for 2-SAT. This behavior also implies a\nnovel type of transient chaos in which the escape rate has an\nexponential-algebraic dependence on the critical parameter \κ \∼\nN^B|\α - \α\χ|1-\γ with 0< \γ < 1. We demonstrate\nthat the transition is generated by the appearance of metastable basins in the\nsolution space as the density of constraints \α is increased.\n