多项式时间中的MAJORITY-3SAT（及相关问题）
MAJORITY-3SAT (and Related Problems) in Polynomial Time
摘要：
多数SAT是确定合取范式（CNF）中的输入$n$变量公式是否至少具有满足赋值的$2^{n-1}$的问题。多数SAT和相关问题已经在各种对概率规划和推理的复杂性感兴趣的AI社区中得到了广泛的研究。虽然Majority-SAT被认为是PP完备的已经有40多年了，但它的一个自然变体的复杂性仍然是开放的：Majority-$k$ SAT，其中输入的CNF公式被限制为最多有$k$的子句宽度。我们证明，对于每一个$k$，多数- $k$ SAT是在P。事实上，对于任何正整数$k$和有理数$ρ\in (0,1)$与有界分母，我们给出了一个算法，可以确定是否一个给定的$k$ -CNF至少有$ρ\cdot 2^n$满足分配，在确定性线性时间（而以前最著名的算法在指数时间运行）。我们的算法有有趣的积极影响计数的复杂性和复杂性的推断，显着降低了已知的复杂性，如E-MAJ- $k$ SAT和MAJ-MAJ- $k$ SAT的相关问题。在我们的方法的核心是一个有效的方法，用于解决阈值计数问题，通过提取向日葵中发现的相应的一个$k$ -CNF集系统。我们还表明，多数- $k$ SAT的易处理性是有点脆弱。对于密切相关的GtMajority-SAT问题（我们问一个给定的公式是否有大于$2^{n-1}$满足分配），这是已知的PP-完全的，我们表明，$k$ SAT是在P的$k\le 3$，但成为NP-完全的$k\geq 4$。这些结果是违反直觉的，因为这些问题的“自然”分类是PP-完全的，而且因为对于所有$k\ge 4$，GtMajority- $k$ SAT和Majority- $k$ SAT的复杂性有明显的差异。
英文摘要：
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 $ρ\in (0,1)$ with bounded denominator, we give an algorithm that can determine whether a given $k$-CNF has at least $ρ\cdot 2^n$ 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 $k\ge 4$.
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。
