随机k-SAT的置信传播引导抽取算法
On belief propagation guided decimation for random k-SAT
摘要：
设F是n个变量m个子句的均匀分布随机k-SAT公式。非构造性的论证表明，F对子句/变量比m/n< r（k）~ 2^kln2是高概率可满足的.然而，没有一个有效的算法是已知的，以找到一个令人满意的分配密度低至m/nr（k），ln（k）/k的非零概率。事实上，密度m/n r（k）.ln（k）/k似乎形成了广泛的局部搜索算法的障碍。极少数似乎能够打破这一障碍的算法之一是一种称为信念传播引导抽取的消息传递算法。它是在深入但不严格的统计力学考虑的基础上提出的。对k= 3，4，5的实验表明，该算法在密度非常接近rk的情况下也能成功.本文首次对BP抽取算法进行了严格的分析，指出当m/n>c.r（k）/k时，当c>0（与k无关）时，BP抽取算法已不能找到满意的分配。
英文摘要：
Let F be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that F is satisfiable for clause/variable ratios m/n< r(k)~2^k ln 2 with high probability. Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m/n r(k).ln(k)/k with a non-vanishing probability. In fact, the density m/n r(k).ln(k)/k seems to form a barrier for a broad class of local search algorithms. One of the very few algorithms that plausibly seemed capable of breaking this barrier is a message passing algorithm called Belief Propagation Guided Decimation. It was put forward on the basis of deep but non-rigorous statistical mechanics considerations. Experiments conducted for k=3,4,5 suggested that the algorithm might succeed for densities very close to r_k. Furnishing the first rigorous analysis of BP decimation, the present paper shows that the algorithm fails to find a satisfying assignment already for m/n>c.r(k)/k, for a constant c>0 (independent of k).
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