随机NAE-K-SAT问题的观测引导抽取算法性能
Performance of the Survey Propagation-guided decimation algorithm for the random NAE-K-SAT problem
摘要：
我们发现，调查的decimation引导抽取算法未能找到令人满意的分配随机实例的“不全相等$K$ -SAT”的问题，如果消息传递迭代的数量是有界的一个常数独立的实例的大小和条款变量的比例是足够大的$K$以上$(1+o_K(1)){2^{K-1}\over K}\log^2 K$。我们的分析实际上适用于广泛的一类算法描述为“顺序本地算法”。这类算法基于局部信息迭代地设置变量，然后在约简实例上递归。调查推理引导的以及信念推理引导的抽取算法-两种广泛研究的基于消息传递的算法，落入这类算法，只要消息传递迭代的次数由常数限制。另一个属于这一类的著名算法是单位子句算法。我们的工作构成了第一个严格的分析性能的SP引导抽取算法。本文的方法基于随机NAE- $K$ -SAT问题解空间的复杂几何结构。我们发现，以上的$(1+o_K(1)){2^{K-1}\over K}\log^2 K$阈值，重叠结构的$m$元组的满意的分配表现出一定的聚类行为的形式表示的约束之间的距离的$m$分配，适当选择的$m$。我们进一步表明，如果一个连续的本地算法成功地找到一个令人满意的分配与概率有界远离零，那么可以构建一个$m$元组的解决方案违反这些限制，从而导致矛盾。随着（引文），这一结果是第一个工作，直接链接的随机约束满足问题的聚类属性的计算困难，找到满意的分配。
英文摘要：
We show that the Survey Propagation-guided decimation algorithm fails to find satisfying assignments on random instances of the "Not-All-Equal-$K$-SAT" problem if the number of message passing iterations is bounded by a constant independent of the size of the instance and the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\log^2 K$ for sufficiently large $K$. Our analysis in fact applies to a broad class of algorithms described as "sequential local algorithms". Such algorithms iteratively set variables based on some local information and then recurse on the reduced instance. Survey Propagation-guided as well as Belief Propagation-guided decimation algorithms - two widely studied message passing based algorithms, fall under this category of algorithms provided the number of message passing iterations is bounded by a constant. Another well-known algorithm falling into this category is the Unit Clause algorithm. Our work constitutes the first rigorous analysis of the performance of the SP-guided decimation algorithm. The approach underlying our paper is based on an intricate geometry of the solution space of random NAE-$K$-SAT problem. We show that above the $(1+o_K(1)){2^{K-1}\over K}\log^2 K$ threshold, the overlap structure of $m$-tuples of satisfying assignments exhibit a certain clustering behavior expressed in the form of constraints on distances between the $m$ assignments, for appropriately chosen $m$. We further show that if a sequential local algorithm succeeds in finding a satisfying assignment with probability bounded away from zero, then one can construct an $m$-tuple of solutions violating these constraints, thus leading to a contradiction. Along with (citation), this result is the first work which directly links the clustering property of random constraint satisfaction problems to the computational hardness of finding satisfying assignments.
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