Given a partially order set (poset) $P$, and a pair of families of ideals $\cI$ and filters $\cF$ in $P$ such that each pair $(I,F)\in \cI\times\cF$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\cF$ and does not contain any member of $\cI$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains $\cA,\cB\subseteq L$ such that no $a\in\cA$ is dominated by any $b\in\cB$, whether $\cA$ and $\cB$ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of $P$, $\cA$ and $\cB$, thus answering an open question in \cite{BK17}. As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.

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关于分配格上的二元化

给定一个偏序集（poset）$P$，以及一对理想族$\cI$ 和$P$ 中的过滤器$\cF$，使得每对$(I,F)\in\cI\times\ cF$ 有一个非空交点，$P$ 上的二元化问题是检查$P$ 中是否存在理想的$X$ 与$\cF$ 的每个成员相交并且不包含$\cI 的任何成员$. 等效地，问题是检查分配格 $L=L(P)$，由其联合不可约集合的偏序 $P$ 和两个给定的反链 $\cA,\cB\subseteq L$ 给出没有 $a\in\cA$ 被任何 $b\in\cB$ 支配，无论 $\cA$ 和 $\cB$ 覆盖（通过支配）整个格子。我们表明该问题可以在 $P$、$\cA$ 和 $\cB$ 大小的拟多项式时间内解决，从而回答了 \cite{BK17} 中的一个悬而未决的问题。作为应用程序，