随机可满足k-CNF公式中满足赋值集的直径
On the diameter of the set of satisfying assignments in random satisfiable k-CNF formulas
摘要：
众所周知，随机k-CNF公式在密度（即子句变量比）约为2 ^k \ln 2时有一个所谓的可满足性阈值：在密度略低于此阈值时，几乎所有k-CNF公式都是可满足的，而在密度略高于此阈值时，几乎没有k-CNF公式是可满足的。在目前的工作中，我们考虑可满足的随机公式，并检查另一个参数-直径的解决方案空间（这是最大的汉明距离之间的一对满意的分配）。以前的研究表明，对于所有密度，直到密度略低于可满足性阈值，直径几乎肯定至少约为n/2（在低得多的密度下为n）。当密度远高于可满足性阈值时，直径几乎肯定为零（一个非常稠密的可满足公式预计只有一个满足的赋值）。在这篇文章中，我们证明了，对于所有的密度高于一个密度，这是略高于可满足性阈值（更准确地说，在比率（1+ \eps）\ln 2，\eps = \eps（k）趋于0，随着k的增长）的直径几乎肯定是O（k2^{-k}n）。这表明，在可满足性阈值附近的密度的相对小的变化（乘法（1 + \eps）因子）使得直径发生显著变化。直径的下降不能归因于大部分公式是不可满足的（因此直径为0），因为我们的条件是公式是可满足的，所以不可满足的公式被排除在考虑之外。
英文摘要：
It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2^k\ln 2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas, and inspect another parameter -- the diameter of the solution space (that is the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly n/2 (and n at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisely at ratio (1+ \eps)2^k \ln 2, \eps=\eps(k) tending to 0 as k grows) the diameter is almost surely O(k2^{-k}n). This shows that a relatively small change in the density around the satisfiability threshold (a multiplicative (1 + \eps) factor), makes a dramatic change in the diameter. This drop in the diameter cannot be attributed to the fact that a larger fraction of the formulas is not satisfiable (and hence have diameter 0), because the non-satisfiable formulas are excluded from consideration by our conditioning that the formula is satisfiable.
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