Efficient constant-factor approximate enumeration of minimal subsets for monotone properties with weight constraints

IF 1 3区 数学 Q3 MATHEMATICS, APPLIED
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引用次数: 0

Abstract

A property Π on a finite set U is monotone if for every XU satisfying Π, every superset YU of X also satisfies Π. Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of U satisfying Π with the minimum weight 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 enumerate multiple small solutions rather than to find a single small solution. To this end, given a weight function w:UQ>0 and kQ>0, we devise algorithms that 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 is at most k and may output some minimal subsets satisfying Π whose weight exceeds k but is at most ck for some constant c1. 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.
带权重约束的单调属性最小子集的高效恒因子近似枚举
如果对于满足 Π 的每个 X⊆U,X 的每个超集 Y⊆U 也满足 Π,那么有限集 U 上的属性 Π 就是单调的。许多组合性质都可以看作是单调性质。以最小权重找到满足 Π 的 U 子集是组合优化的核心问题。虽然已经开发出了许多近似/精确算法来解决这类问题,但由于很难针对实际问题建立精确的数学模型,这些算法得到的解决方案往往不适合实际应用。克服这一困难的一种可行方法是枚举多个小解决方案,而不是寻找单一的小解决方案。为此,给定一个权重函数 w:U→Q>0 和 k∈Q>0,我们设计了一些算法,可以近似枚举权重至多为 k、满足各种单调属性 Π 的 U 的所有最小子集,这里的 "近似枚举 "是指算法输出所有满足 Π 的最小子集,其权重至多为 k,并且可能输出一些满足 Π 的最小子集,其权重超过 k,但在某个常数 c≥1 时至多为 ck。通过这些算法,我们可以高效地枚举最小顶点覆盖、有界度图中的最小支配集、最小反馈顶点集、有界秩超图中的最小命中集等、权重最多为 k,且近似系数恒定。
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来源期刊
Discrete Applied Mathematics
Discrete Applied Mathematics 数学-应用数学
CiteScore
2.30
自引率
9.10%
发文量
422
审稿时长
4.5 months
期刊介绍: The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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