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arXiv 2009.08830cs.DS
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具有权重约束的单调性质极小子集的高效常数因子近似枚举

Efficient Constant-Factor Approximate Enumeration of Minimal Subsets for Monotone Properties with Weight Constraints

Yasuaki Kobayashi, Kazuhiro Kurita, Kunihiro Wasa

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AI总结：

针对单调性质下权重受限的极小子集枚举问题，提出常数因子近似枚举算法，可高效处理顶点覆盖、支配集、反馈顶点集和击中集等组合对象。

摘要：

有限集合 $U$ 上的性质 $Π$ 被称为单调的，条件是：对于每个满足 $Π$ 的 $X \subseteq U$，$X$ 的每个超集 $Y \subseteq U$ 也满足 $Π$。许多组合性质都可以视为单调性质。寻找 $U$ 中满足 $Π$ 的最小子集是组合优化中的核心问题。尽管人们已经开发了许多近似/精确算法来求解众多性质上的此类问题，但由于难以针对现实问题建立准确的数学模型，这些算法得到的单个解往往不适用于现实应用。克服这一困难的一种有前景的方法是枚举多个小解，而不是只寻找一个小解。为此，在给定权重函数 $w: U \to \mathbb N$ 和整数 $k$ 的情况下，我们设计了若干算法，针对各种单调性质 $Π$，近似枚举 $U$ 中权重不超过 $k$ 且满足 $Π$ 的所有极小子集；这里的“近似枚举”是指算法输出权重不超过 $k$ 且满足 $Π$ 的所有极小子集，并且也可能输出一些权重超过 $k$、但对于某个常数 $c \ge 1$ 至多为 $ck$ 的满足 $Π$ 的极小子集。这些算法使我们能够以常数近似因子高效枚举权重不超过 $k$ 的极小顶点覆盖、有界度数图中的极小支配集、极小反馈顶点集以及有界秩超图中的极小击中集等。

英文摘要：

A property $Π$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $Π$, every superset $Y \subseteq U$ of $X$ also satisfies $Π$. Many combinatorial properties can be seen as monotone properties. The problem of finding a minimum subset of $U$ satisfying $Π$ is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to \emph{enumerate} multiple small solutions rather than to \emph{find} a single small solution. To this end, given a weight function $w: U \to \mathbb N$ and an integer $k$, we devise algorithms that \emph{approximately} enumerate all minimal subsets of $U$ with weight at most $k$ satisfying $Π$ for various monotone properties $Π$, where "approximate enumeration" means that algorithms output all minimal subsets satisfying $Π$ whose weight at most $k$ and may output some minimal subsets satisfying $Π$ whose weight exceeds $k$ but is at most $ck$ for some constant $c \ge 1$. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most $k$ with constant approximation factors.

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