种植SAT上局部搜索的相变
Phase transition for Local Search on planted SAT
摘要：
局部搜索算法（或爬山法，或迭代改进法）是解决可满足性和最大可满足性问题的最简单的算法之一。它是许多可满足性和最大可满足性求解器的一部分，用于为更复杂的算法找到一个好的起点，并改进候选解。本文分析了随机种植3-CNF公式的局部搜索问题。我们证明了如果k<7/6使得子句与变量的比率小于kln（n）（n是CNF中变量的数量），那么局部搜索whp找不到满意的分配，如果k>7/6使得子句与变量的比率大于kln（n）$，那么局部搜索whp找到满意的分配.作为一个副产品，我们还表明，对于任何常数r有g，使本地搜索应用到一个随机的（不一定种植）3-CNF与条款变量比r产生的分配，满足至少gn条款小于最大数量的可满足的条款。
英文摘要：
The Local Search algorithm (or Hill Climbing, or Iterative Improvement) is one of the simplest heuristics to solve the Satisfiability and Max-Satisfiability problems. It is a part of many satisfiability and max-satisfiability solvers, where it is used to find a good starting point for a more sophisticated heuristics, and to improve a candidate solution. In this paper we give an analysis of Local Search on random planted 3-CNF formulas. We show that if there is k<7/6 such that the clause-to-variable ratio is less than k ln(n) (n is the number of variables in a CNF) then Local Search whp does not find a satisfying assignment, and if there is k>7/6 such that the clause-to-variable ratio is greater than k ln(n)$ then the local search whp finds a satisfying assignment. As a byproduct we also show that for any constant r there is g such that Local Search applied to a random (not necessarily planted) 3-CNF with clause-to-variable ratio r produces an assignment that satisfies at least gn clauses less than the maximal number of satisfiable clauses.
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