PPSZ, for long time the fastest known algorithm for $k$-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to $0$ or $1$, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being implied by a small set of clauses). We show that PPSZ performs exponentially better than previously known, for all $k \geq 3$. For Unique-$3$-SAT we bound its running time by $O(1.306973^{n})$, which is somewhat better than the algorithm of Hansen, Kaplan, Zamir, and Zwick, which runs in time $O(1.306995^n)$. Before that, the best known upper bound for Unique-$3$-SAT was $O(1.3070319^n)$. All improvements are achieved without changing the original PPSZ. The core idea is to pretend that PPSZ does not process the variables in uniformly random order, but according to a carefully designed distribution. We write "pretend" since this can be done without any actual change to the algorithm.
翻译:PPSZ是长期以来已知的美元-SAT的最快算法,它以随机顺序通过输入公式的变量进行计算;然后,每个变量随机地设定为0美元或1美元,除非正确值可以通过高效执行规则(如小宽分辨率,或由一组小条款暗含)推断出来。我们显示,PPSZ在全部美元/克(3美元)方面表现的指数比先前已知的要好。对于Unique-3美元,我们将其运行时间约束在O(1.306973 ⁇ n)美元上,这比Hansen、Kaplan、Zamir和Zwick的算法略好一些,后者的算法运行时间为0.306995美元。在此之前,已知的Unique-美元-美元-SAT的上限是$(1.3070319元)美元。所有改进都是在不改变原PPSZ的情况下实现的。核心想法是假装PPSZ不按统一的随机顺序处理变量,而是按照仔细设计的分布进行。我们写“prependendend ” 之后, 任何这种算出“ pretendendendendendendend ” 。