RB模型实例集上置信传播算法的收敛性
Convergence of the Belief Propagation Algorithm for RB Model Instances

DOI：10.13328/j.cnki.jos.004877

 作者 单位 E-mail 王晓峰 北方民族大学 计算机科学系, 宁夏 银川 750021贵州大学 计算机科学系, 贵州 贵阳 550025 许道云 贵州大学 计算机科学系, 贵州 贵阳 550025 dyxu@gzu.edu.cn

置信传播算法求解RBk，n，α，rcp）模型实例时非常有效，几乎能够有效求解接近可满足性相变点的难解实例.然而，因子图带有回路的实例，置信传播算法不总有效，常表现为不收敛.对于这种现象，至今缺少系统的理论解释.置信传播算法是最为基础的信息传播算法，对置信传播算法的收敛性分析是其他信息传播算法收敛性分析的重要基础.在RBk，n，α，rcp）模型中，取k=2，α>（1/k），rc>0均为常数，且满足ke-（α/（rc））≥1.证明了如果p∈（0，n-2α），则置信传播算法在RBk，n，α，rcp）模型产生的随机实例集上高概率收敛.最后，在RBk，n，α，rcp）模型上选取了几组不同的数据进行数值模拟，实验结果表明该结论有效.当问题规模n增大时，在RBk，n，α，rcp）模型的可满足区域，实验收敛区间趋于一个固定范围，而理论收敛区间逐渐变窄.原因在于，RBk，n，α，rcp）模型是一个具有增长定义域的随机CSP实例产生模型，不协调赋值的数目与参数p及问题规模n有关.

Belief propagation algorithm is very effective in finding satisfying assignments for RB(k,n,α,rc,p) model instances where hard region becomes narrower. However, belief propagation algorithm does not always converge for factor graphs with cycles. Unfortunately, rigorous theoretical proof of this phenomenon is still lacking. Belief propagation algorithm is the most basic message passing algorithms. This study derives the convergence conditions of the belief propagation algorithm for solving RB(k,n,α,rc,p) model instances. In the RB(k,n,α,rc,p) model with k=2, α>（1/k）， rc>0 and which proves that BP will be converged with high probability if p∈(0,n-2α). Experimental results show that such convergence conditions of belief propagation algorithm are very effective in two different group data from the random RB(k,n,α,rc,p) model instances. In the RB(k,n,α,rc,p) model, when n increases, the experimental convergence interval is fixed range, while the theory convergence interval become narrower. It is because the number of incompatible assignment are determined by n and p in the RB(k,n,α,rc,p).
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