发表机构
Hokkaido University; Okayama University; Hosei University（北海道大学; 冈山大学; 法政大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对长期悬而未决的单调布尔函数对偶化问题，该文基于指数时间假设（ETH），通过将3SAT问题亚指数时间归约到其补问题，证明了对偶化问题不存在拟多项式时间算法，进而否定了其输出多项式时间可解性。
AI 中文摘要
单调布尔函数的对偶化（等价于超图中极小横截的枚举）是一个长期存在的问题，其输出多项式时间可解性仍未解决。尽管各类特殊情形已得到广泛研究，但Fredman和Khachiyan提出的一般情形最优算法运行时间为拟多项式。本文给出从\textsc{3SAT}到\textsc{Dual}补问题的亚指数时间归约：给定一个含$n$变量的3CNF公式，该归约构造总规模为$2^{\bigoh(n^{2/3}(\log n)^{1/3})}$的超图$\mathcal H$和$\mathcal L$，使得$\mathcal L \subseteq \Tr(\mathcal H)$，且该公式可满足当且仅当$\mathcal L\neq \Tr(\mathcal H)$。作为该归约的推论，假设指数时间假设（ETH）成立，则\textsc{Dual}和\textsc{Dualization}均不存在运行时间为$N^{o(\sqrt{\log N/\log\log N})}$的算法，其中$N$是\textsc{Dual}的输入规模，也是\textsc{Dualization}的输入与输出总规模。特别地，在ETH下\textsc{Dualization}无法在输出多项式时间内求解。
英文摘要
Dualizing monotone Boolean functions (or equivalently, enumerating minimal transversals in hypergraphs) is a long-standing problem whose output-polynomial-time solvability remains open. While various special cases have been extensively studied, the state-of-the-art algorithm for the general case, due to Fredman and Khachiyan, runs in quasipolynomial time. This paper presents a subexponential-time reduction from \textsc{3SAT} to the complement of \textsc{Dual}: Given a 3CNF formula with $n$ variables, the reduction constructs hypergraphs $\mathcal H$ and $\mathcal L$ of total size $2^{\bigoh(n^{2/3}(\log n)^{1/3})}$ such that $\mathcal L \subseteq \Tr(\mathcal H)$ and the formula is satisfiable if and only if $\mathcal L\neq \Tr(\mathcal H)$. As a consequence of this reduction, assuming the Exponential Time Hypothesis (ETH), neither \textsc{Dual} nor \textsc{Dualization} admits an algorithm running in $N^{o(\sqrt{\log N/\log\log N})}$ time, where $N$ is the input size for \textsc{Dual} and the combined input and output size for \textsc{Dualization}. In particular, \textsc{Dualization} cannot be solved in output-polynomial time under ETH.
