定义部分布尔函数的一对形式的不可逼近性
Inapproximability of a Pair of Forms Defining a Partial Boolean Function
摘要：
考虑两个布尔函数$f, g \colon \{0,1\}^J \to \{0,1\}$的联合极小化形式问题，使得$f + g \leq 1$和从而分离不相交的集合$A \cup B \subseteq \{0,1\}^J$，使得$f(A) = \{1\}$和$g(B) = \{1\}$ .我们假设，这个问题是更容易解决或近似比众所周知的问题，最小化一个布尔函数$h: \{0,1\}^J \to \{0,1\}$的形式，使$h(A) = \{1\}$和$h(B) = \{0\}$。对于一个大类的形式，包括二叉决策树和有序二叉决策图，我们反驳了这一假设。对于析取范式，我们证明了这个问题至少和MIN-SET-COVER一样难。对于所有这些形式，我们建立没有$o(\ln (|A| + |B| -1))$ -近似算法存在，除非P $=$ NP。
英文摘要：
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.
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