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A novel weighting scheme for random k-SAT

关于随机 k-SAT 的新加权方法

Abstract

Consider a random k-conjunctive normal form F k (n, rn) with n variables and rn clauses. We prove that if the probability that the formula F k (n, rn) is satisfiable tends to 0 as n→∞, then r ⩾ 2.83, 8.09, 18.91, 40.81, and 84.87, for k = 3, 4, 5, 6, and 7, respectively.

摘要

创新点

SAT 问题是第一个被证明的 NP 完全问题, 是理论计算机科学研究的核心问题之一。 自从上世纪九十年代初发现 NP 完全问题存在相变现象以来, 随机 k-SAT 问题的相变现象受到了理论计算机科学、 人工智能、组合学、 概率和统计物理等领域的广泛关注。 其中人们最感兴趣的是关于随机 k-SAT 问题的可满足性阈值猜想。 2014 年, Ding Jian 等人证明了当 k 充分大时可满足性阈值猜想成立, 但当 k 比较小时的进展仍然是比较缓慢。 在本文中, 我们充分挖掘加权二阶矩方法的潜力, 得到了随机 k-SAT 阈值的下界: k = 3, 4, 5, 6, 7 时, 分别为 2.83, 8.09, 18.91, 40.81 和 84.87。 其中 18.91, 40.81 和 84.87 是目前已知最好的下界。

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Correspondence to Ke Xu.

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Liu, J., Xu, K. A novel weighting scheme for random k-SAT. Sci. China Inf. Sci. 59, 92101 (2016). https://doi.org/10.1007/s11432-016-5526-8

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Keywords

  • complexity
  • satisfiability
  • phase transition
  • second moment method
  • weighting scheme

关键词

  • 计算复杂性
  • 可满足性
  • 相变
  • 二阶矩
  • 加权方法