随机布尔CSP的满足赋值：簇与重叠
Satisfying assignments of Random Boolean CSP: Clusters and Overlaps
摘要：
随机CSP的解的重叠分布是其解空间的整体几何的指示器。对于随机$k$ -SAT，统计物理学的非严格方法支持“一步复制对称破缺”方法的有效性。其中一些预测在\cite{cond-mat/0504070/prl} \cite{cond-mat/0506053}中得到了严格的证实。证明了随机$k$的重叠分布$k\geq 9$具有不连续支撑.此外，Achlioptas和Ricci-Tersenghi证明，对于随机$k$，$k\geq 8$。当约束密度足够接近相变点时，满足赋值的簇的个数为指数级，且不同簇中满足赋值之间的距离为线性.我们的目标是了解随机CSP的结构特性，导致解决方案聚类。为此，我们证明了两个结果的集群结构的解决方案的随机模型下，从Molloy（STOC 2002）1。对于所有约束集$S$（在Creamou和Daude（2004），Istrate（2005）中明确描述），$SAT(S)$具有一个尖锐的阈值，并且所有的$q\in (0,1]$、$q$ -重叠- $SAT(S)$都具有一个尖锐的阈值（即，Mora等人的方法的第一步在所有非平凡情况下都有效）。2.对于任何约束密度值$c<1$，2-SAT形式的随机实例w. h. p.，一个单一的集群。而且，对于和任何$q\in (0,1]$，这样的实例具有w. h. p.两个满足重叠分配的$\sim q$。因此，正如统计物理学预测所预期的那样，Mora等人的方法的第二步对于2-SAT失败了。
英文摘要：
The distribution of overlaps of solutions of a random CSP is an indicator of the overall geometry of its solution space. For random $k$-SAT, nonrigorous methods from Statistical Physics support the validity of the ``one step replica symmetry breaking'' approach. Some of these predictions were rigorously confirmed in \cite{cond-mat/0504070/prl} \cite{cond-mat/0506053}. There it is proved that the overlap distribution of random $k$-SAT, $k\geq 9$, has discontinuous support. Furthermore, Achlioptas and Ricci-Tersenghi proved that, for random $k$-SAT, $k\geq 8$. and constraint densities close enough to the phase transition there exists an exponential number of clusters of satisfying assignments; moreover, the distance between satisfying assignments in different clusters is linear. We aim to understand the structural properties of random CSP that lead to solution clustering. To this end, we prove two results on the cluster structure of solutions for binary CSP under the random model from Molloy (STOC 2002) 1. For all constraint sets $S$ (described explicitly in Creignou and Daude (2004), Istrate (2005)) s.t. $SAT(S)$ has a sharp threshold and all $q\in (0,1]$, $q$-overlap-$SAT(S)$ has a sharp threshold (i.e. the first step of the approach in Mora et al. works in all nontrivial cases). 2. For any constraint density value $c<1$, the set of solutions of a random instance of 2-SAT form, w.h.p., a single cluster. Also, for and any $q\in (0,1]$ such an instance has w.h.p. two satisfying assignment of overlap $\sim q$. Thus, as expected from Statistical Physics predictions, the second step of the approach in Mora et al. fails for 2-SAT.
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