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 ∼ 10 2 . 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.

The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping utt + ∆ 2 u − m(∇u 2)∆u + ut p ut + f (u) = h, in Ω × R + , p ≥ 0 on a bounded smooth domain Ω ⊂ R n with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient m(∇u 2) and the nonlinear term f (u), the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.

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We study solution-space structure and solution-finding algorithms of a representative hard random constraint satisfaction problem with growing domains known as Model RB. Using rigorous methods, we show that solutions are grouped into disconnected clusters before the theoretical satisfiability phase transition point. Using the cavity method, it is shown that the entropy density obtained by belief propagation (BP) on random Model RB instances, which corresponds well to the analytical results, vanishes as the control parameter (constraint tightness) approaches the satisfiability threshold. From an algorithmic point of view, we find that reinforced BP, which performs much better than all existing algorithms, allows us to find solutions efficiently for instances in the regime that is very close to the satisfiability transition. These results also can shed light on the effectiveness of BP reinforcement on problems with a large number of states.

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