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 10². 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.

我们提出了三种使用异步更新策略的信念传播（BP）引导抽取算法来解决具有增长域的随机约束满足问题的原型，称为模型 RB。对于模型 RB，已经严格建立了精确的可满足性相变，并且几乎所有实例都在过渡区域内是固有硬的。寻找模型 RB 的随机实例的解决方案非常具有挑战性，问题大小限制为 10 ². 我们提出的BP引导抽取算法分别称为异步更新置信传播（ABP）算法、异步更新置信传播*（ABP*）算法和变阶异步更新置信传播（VABP）算法。在算法的 BP 部分，我们采用异步更新策略来获取约束和变量之间的最新传递消息，这可以提高 BP 方程的收敛性。我们还使用了一个阻尼因子，将具有一定权重的旧消息添加到从变量发送到约束的新消息中，以减少 BP 方程收敛过程中振荡的发生。在ABP算法中，我们根据BP方程收敛后得到的消息计算所有变量的边际概率分布，然后选择偏差最大的变量并将其值固定在具有最大概率的组件上。而 ABP* 算法则考虑如果 BP 方程不收敛，如何继续抽取过程。与前两种算法不同的是，在VABP算法中，我们首先选择变量的一个随机顺序，然后在BP收敛后按照给定的顺序给变量赋值。实验结果表明，即使约束紧度接近理论可满足性阈值，这三种 BP 引导抽取算法在求解模型 RB 的随机实例方面似乎非常有效。为了评估 ABP 算法的性能，我们还提供了同步更新 BP 算法作为比较。还讨论了每个时间步长所选变量的熵和变量在不同约束松紧度下的平均自由度。此外，我们分析了BP方程的收敛性以及BP引导抽取算法抽取过程中所选变量的阶数的影响。