调查传播及其推广的新视角
A New Look at Survey Propagation and its Generalizations
摘要：
本文提供了一个新的概念的角度调查传播，这是一个迭代算法最近推出的统计物理界，是非常有效的解决随机k-SAT问题，即使密度接近可满足性阈值。我们首先描述了任何SAT公式可以与一个新的家庭的马尔可夫随机场（MRF），参数化的实数ρ \in [0，1]。然后，我们表明，应用信念传播-一个众所周知的“消息传递”技术估计边际概率-这一系列的MRF恢复一个已知的家庭的算法，从一个极端的纯调查传播（ρ = 1），以标准的信念传播在SAT分配的均匀分布在另一个极端（ρ = 0）。在这些MRF中的可满足性有一个自然的解释为部分可满足性分配，在其上可以定义一个偏序。我们将核心隔离为这个偏序中的最小元素，它们也是调查传播的不动点，并且是ρ = 1时MRF中唯一具有正概率的分配。我们的实验结果为k = 3的随机公式的解决方案通常不具有非平凡的核心。这使得有必要研究ρ <1的部分赋值空间的结构，并研究非常接近核的赋值的作用。为此，我们研究了相关的格结构，并证明了一个保权恒等式，该恒等式表明，任何ρ> 0的MRF都可以被视为满足分配（ρ = 0）的均匀分布的"平滑"版本。最后，我们隔离的吉布斯采样和消息传递算法，是典型的K-SAT问题的集合的属性。
英文摘要：
This paper provides a new conceptual perspective on survey propagation, which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random k-SAT problems even with densities close to the satisfiability threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number ρ\in [0,1]. We then show that applying belief propagation--a well-known ``message-passing'' technique for estimating marginal probabilities--to this family of MRFs recovers a known family of algorithms, ranging from pure survey propagation at one extreme (ρ= 1) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme (ρ= 0). Configurations in these MRFs have a natural interpretation as partial satisfiability assignments, on which a partial order can be defined. We isolate cores as minimal elements in this partial ordering, which are also fixed points of survey propagation and the only assignments with positive probability in the MRF for ρ=1. Our experimental results for k=3 suggest that solutions of random formulas typically do not possess non-trivial cores. This makes it necessary to study the structure of the space of partial assignments for ρ<1 and investigate the role of assignments that are very close to being cores. To that end, we investigate the associated lattice structure, and prove a weight-preserving identity that shows how any MRF with ρ>0 can be viewed as a ``smoothed'' version of the uniform distribution over satisfying assignments (ρ=0). Finally, we isolate properties of Gibbs sampling and message-passing algorithms that are typical for an ensemble of k-SAT problems.
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