A message-passing approach to random constraint satisfaction problems with growing domains

Message-passing algorithms based on belief propagation (BP) are implemented on a random constraint satisfaction problem (CSP) referred to as model RB, which is a prototype of hard random CSPs with growing domain size. In model RB, the number of candidate discrete values (the domain size) of each variable increases polynomially with the variable number N of the problem formula. Although the satisfiability threshold of model RB is exactly known, finding solutions for a single problem formula is quite challenging and attempts have been limited to cases of N ∼ 10². In this paper, we propose two different kinds of message-passing algorithms guided by BP for this problem. Numerical simulations demonstrate that these algorithms allow us to find a solution for random formulas of model RB with constraint tightness slightly less than p_cr, the threshold value for the satisfiability phase transition. To evaluate the performance of these algorithms, we also provide a local search algorithm (random walk) as a comparison. Besides this, the simulated time dependence of the problem size N and the entropy of the variables for growing domain size are discussed.