Optimization of the dynamic transition in the continuous coloring problem

Abstract Random constraint satisfaction problems (CSPs) can exhibit a phase where the number of constraints per variable α makes system solvable in theory on one hand, but also search for solution hard, meaning that common algorithms such as Monte Carlo (MC) method fail to find solution. The onset this hardness is deeply linked appearance dynamical transition space problem breaks into an exponential clusters. exact position not universal with respect details Hamiltonian chooses represent given problem. In paper, we develop some theoretical tools order systematic way build maximizes dynamic d threshold. To illustrate our techniques, will concentrate continuous coloring, tries set angle x i ∈ [0; 2 π ] each node network no adjacent nodes are closer than threshold θ , cos( − j )⩽ cos . This be both seen version discrete graph coloring or one-dimensional Mari–Krzakala–Kurchan model. relevance model stems from fact CSPs sparse random graphs remain largely unexplored statistical physics. We show sufficiently small presents first and compute dynamical, condensation Kesten–Stigum transitions; compare analytical predictions MC simulations values = / q q ∈ N Choosing allows us easily results renowned coloring.