Belief propagation guided decimation algorithms for random constraint satisfaction problems with growing domains

We propose three kinds of belief propagation (BP) guided decimation algorithms using asynchronous updating strategy to solve a prototype of random constraint satisfaction problem with growing domains referred to as model RB. For model RB, the exact satisfiability phase transitions have been established rigorously, and almost all instances are intrinsic hard in the transition region. Finding solutions of a random instance of model RB is very challenging, and the problem size is limited to 10(2). The BP guided decimation algorithms we proposed are called asynchronous updating belief propagation (ABP) algorithm, asynchronous updating belief propagation* (ABP*) algorithm, and asynchronous updating belief propagation with variable order (VABP) algorithm, respectively. In the BP part of the algorithms, we adopt asynchronous updating strategy to obtain the latest passing messages between constraints and variables, which can improve the convergence of BP equations. We also use a damping factor that adds the old messages with a certain weight into the new messages sent from variables to constraints, to reduce the occurrence of oscillation during the convergence of BP equations. In the ABP algorithm, we compute the marginal probability distribution of all variables according to the messages obtained after the BP equations converge, then select the most biased variable and fix its value on the component with the maximum probability. While the ABP* algorithm considers how to continue the decimation process if the BP equations do not converge. Different from the previous two algorithms, in the VABP algorithm, we first choose a random order of the variables, and then assign values to the variables according to the given order after BP converges. Experimental results suggest that the three kinds of BP guided decimation algorithms appear to be very effective in solving random instances of model RB even when the constraint tightness is close to the theoretical satisfiability threshold. To evaluate the performance of the ABP algorithm, we also provide synchronous updating BP algorithms as a comparison. The entropy of the selected variable at each time step and the average freedom of the variables at different constraint tightness are also discussed. Besides, we analyze the convergence of BP equations and the influence of the order of the selected variables in the decimation process of the BP guided decimation algorithms.