Majority-SAT is the problem of determining whether an input $n$-variable formula in conjunctive normal form (CNF) has at least $2^{n-1}$ satisfying assignments. Majority-SAT and related problems have been studied extensively in various AI communities interested in the complexity of probabilistic planning and inference. Although Majority-SAT has been known to be PP-complete for over 40 years, the complexity of a natural variant has remained open: Majority-$k$SAT, where the input CNF formula is restricted to have clause width at most $k$. We prove that for every $k$, Majority-$k$SAT is in P. In fact, for any positive integer $k$ and rational $\rho \in (0,1)$ with bounded denominator, we give an algorithm that can determine whether a given $k$-CNF has at least $\rho \cdot 2^n$ satisfying assignments, in deterministic linear time (whereas the previous best-known algorithm ran in exponential time). Our algorithms have interesting positive implications for counting complexity and the complexity of inference, significantly reducing the known complexities of related problems such as E-MAJ-$k$SAT and MAJ-MAJ-$k$SAT. At the heart of our approach is an efficient method for solving threshold counting problems by extracting sunflowers found in the corresponding set system of a $k$-CNF. We also show that the tractability of Majority-$k$SAT is somewhat fragile. For the closely related GtMajority-SAT problem (where we ask whether a given formula has greater than $2^{n-1}$ satisfying assignments) which is known to be PP-complete, we show that GtMajority-$k$SAT is in P for $k\le 3$, but becomes NP-complete for $k\geq 4$. These results are counterintuitive, because the ``natural'' classifications of these problems would have been PP-completeness, and because there is a stark difference in the complexity of GtMajority-$k$SAT and Majority-$k$SAT for all $k\ge 4$.
翻译:多数卫星是一个问题,它决定一个投入值为美元、可变公式的普通形式(CNF)是否至少有2美元、可变公式($1美元)、满意的任务。 多数卫星及相关问题已经在对概率规划和推断的复杂性感兴趣的各个AI社区中得到了广泛的研究。 虽然多数卫星已经知道至少完成PP-40多年,但自然变异的复杂性仍然是开放的: 多数-美元($美元),输入的CNF公式只能有最易变的发价宽度。我们证明,每1美元、Mostity-1美元、可变值美元,P.事实上,对于任何正整美元和合理的美元(0.1美元),具有闭塞分度的美元,我们给出的算法可以确定一个给定的美元-CNF至少为$2美元,但是在确定性的直线性任务中(我们之前最著名的算法是指数时间 ) 。我们的算法对于精确的复杂程度和精确的IMA 直径的算法是否都显示我们所知道的IMA的方法。