针对若干最小成本图问题的好坏3/2近似算法的简化分析
A Simplified Analysis of the Good-Bad $3/2$-Approximation Algorithm for Some Minimum-Cost Graph Problems
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- Cornell University, School of Operations Research and Information Engineering（康奈尔大学运筹学与信息工程学院）
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中文总结 AI 辅助
本文针对下单调函数，对可实现3/2近似比的好坏算法开展简化分析，该算法可用于求解最小成本边集相关图问题。
中文摘要 AI 辅助
本文研究由Couëtoux提出的一种简单贪心近似算法——好坏算法，用于寻找最小成本边集，使得每个连通分量至少包含k个顶点。Couëtoux证明该算法对该问题达到3/2近似比。Davis与Williamson将此结果扩展至更一般问题：寻找最小成本边集，使其从每个满足h(S)=1的割集S⊆V中至少包含一条边，其中h:2^V→{0,1}为下单调函数，即若h(S)=1，则对所有非空子集T⊆S有h(T)=1。原问题对应|S|<k时h(S)=1的情况。本文针对下单调函数给出好坏算法的简化分析。
英文摘要
In this paper, we consider an easy greedy approximation algorithm, the good-bad algorithm, introduced by Couëtoux for finding a minimum-cost set of edges such that every connected component has at least $k$ vertices. Couëtoux proves that the good-bad algorithm achieves a $3/2$-approximation for this problem. Davis and Williamson extend this result to the more general problem of finding a minimum-cost edge set that contains at least one edge from every cut $S\subseteq V$ satisfying $h(S) = 1$ where $h:2^V \rightarrow \{0,1\}$ is downward monotone; that is, $h(S) = 1$ implies $h(T) = 1$ for every nonempty subset $T \subseteq S$. The original problem corresponds to $h(S) =1$ when $|S|<k$. We give a simplified analysis of the good-bad algorithm for downward monotone functions.
