arXiv math/00051362000-05math.PR

关于随机k-SAT的临界指数

On the critical exponents of random k-SAT

David B. Wilson

备注：: 11 pages. v2 has revised introduction and updated references
发表信息：: Random Structures and Algorithms, 21(2):182--195, 2002
DOI：: 10.1002/rsa.10050

摘要：

随机布尔公式的可满足性问题近来引起了人们的极大兴趣。随机k-SAT公式是m个随机子句的合取，每个子句是k个文字（变量或其否定）的析取。众所周知，当变量数n很大时，从可满足到不可满足会有一个急剧的转变;在2-SAT的情况下，当m/n> 1时会发生这种情况，对于3-SAT，临界比率被认为是m/n ~ 4.2。这种转变的尖锐性由临界指数表征，有时称为ν=ν_k（ν的值越小，转变越尖锐）。实验表明，ν_3 = 1.5+-0.1，ν_4 = 1.25+-0.05，ν_5=1.1+-0.05，ν_6 = 1.05+-0.05，而物理学则认为，当k -->无穷大时，ν_k --> 1。我们在这里给出一个简单的证明，证明这些指数中的每一个至少是2（假设指数是定义良好的）。这一结果适用于随机k-SAT公式的三个标准集合中的每一个：m个子句随机一致选择而不替换，m个子句随机一致选择而替换，每个子句以概率p独立于其他子句选择。我们也得到了类似的结果的q-可染性和随机图中的q-核心的外观。

英文摘要：

There has been much recent interest in the satisfiability of random Boolean formulas. A random k-SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called ν=ν_k (the smaller the value of νthe sharper the transition). Experiments have suggested that ν_3 = 1.5+-0.1, ν_4 = 1.25+-0.05, ν_5=1.1+-0.05, ν_6 = 1.05+-0.05, and heuristics have suggested that ν_k --> 1 as k --> infinity. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well-defined). This result holds for each of the three standard ensembles of random k-SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q-colorability and the appearance of a q-core in a random graph.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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