We consider the problem of jointly minimizing forms of two Boolean functions $f, g \colon \{0,1\}^J \to \{0,1\}$ such that $f + g \leq 1$ and so as to separate disjoint sets $A \cup B \subseteq \{0,1\}^J$ such that $f(A) = \{1\}$ and $g(B) = \{1\}$. We hypothesize that this problem is easier to solve or approximate than the well-understood problem of minimizing the form of one Boolean function $h: \{0,1\}^J \to \{0,1\}$ such that $h(A) = \{1\}$ and $h(B) = \{0\}$. For a large class of forms, including binary decision trees and ordered binary decision diagrams, we refute this hypothesis. For disjunctive normal forms, we show that the problem is at least as hard as MIN-SET-COVER. For all these forms, we establish that no $o(\ln (|A| + |B| -1))$-approximation algorithm exists unless P$=$NP.
翻译：我们考虑的是共同尽量减少两种布尔函数的形式问题,例如,g 克诺 = 0.1 ⁇ J = 0.1 ⁇ J = 0.1 ⁇ $,例如,美元+ g = leq 1美元,并且为了分离不连接设置 $A\ cup B = subseteq = 0.1 ⁇ ⁇ J 美元,例如,美元(A) = ⁇ 1 ⁇ $ 和 美元(B) = ⁇ 1 + ⁇ 1 ⁇ $。我们假想,这个问题比尽可能减少一个布尔函数形式的深奥问题更容易解决或接近 $h : 0.1 J + + + 0.1 美元,因此,对于大类形式,包括二进制决定树和订购的二进制决定图,我们反驳了这一假设。关于不相容的正常形式,我们说,问题至少像 MIN-SET-CLO一样难以解决。关于所有这些形式的问题,我们确定,除非存在 $NP-MQ___Q____Q_Q___Q________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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