DAG中最大叶子生成树形图问题的4/3近似算法
A $\frac{4}{3}$-Approximation for the Maximum Leaf Spanning Arborescence Problem in DAGs
AI总结：
本文通过为遗传3-集合打包问题设计基于局部搜索的4/3近似算法，并利用Fernandes和Lintzmayer的归约框架，首次将DAG中最大叶子生成树形图问题的近似比从7/5改进到4/3。
摘要：
最大叶子生成树形图问题（MLSA）定义如下：给定一个有向图$G$和一个顶点$r\in V(G)$，从$r$出发可以到达所有其他顶点，要求找到一棵以$r$为根的生成树形图，使得叶子数（出度为零的顶点）最大化。MLSA在广播领域有应用，其中消息需要从源顶点沿给定网络中树形图的弧传输到所有其他顶点。在此过程中，希望尽可能多的顶点只需要接收而不需要转发消息，因为这类顶点的构建成本天然更低。我们研究MLSA的多项式时间近似算法。对于一般有向图，目前最先进的是$\min\{\sqrt{\mathrm{OPT}},92\}$近似。在输入图无环的特殊情形（仍然是APX困难的）中，已知最好的近似保证$\frac{7}{5}$由Fernandes和Lintzmayer给出：他们证明，对于遗传3-集合打包问题（加权3-集合打包的一个特例）的任何$\alpha$近似，都能为无环有向图（DAG）中的MLSA导出$\max\{\frac{4}{3},\alpha\}$近似，并为遗传3-集合打包问题提供了一个$\frac{7}{5}$近似。在本文中，我们为遗传3-集合打包问题获得了$\frac{4}{3}$近似，从而也为DAG中的MLSA获得了$\frac{4}{3}$近似。在此过程中，我们设法充分发挥了Fernandes和Lintzmayer所给归约的全部潜力。我们研究的算法是一个简单的局部搜索过程，考虑大小不超过$10$的交换。其分析依赖于一个两阶段记账论证。
英文摘要：
The Maximum Leaf Spanning Arborescence problem (MLSA) is defined as follows: Given a directed graph $G$ and a vertex $r\in V(G)$ from which every other vertex is reachable, find a spanning arborescence rooted at $r$ maximizing the number of leaves (vertices with out-degree zero). The MLSA has applications in broadcasting, where a message needs to be transferred from a source vertex to all other vertices along the arcs of an arborescence in a given network. In doing so, it is desirable to have as many vertices as possible that only need to receive, but not pass on messages since they are inherently cheaper to build. We study polynomial-time approximation algorithms for the MLSA. For general digraphs, the state-of-the-art is a $\min\{\sqrt{\mathrm{OPT}},92\}$-approximation. In the (still APX-hard) special case where the input graph is acyclic, the best known approximation guarantee of $\frac{7}{5}$ is due to Fernandes and Lintzmayer: They prove that any $α$-approximation for the \emph{hereditary $3$-set packing problem}, a special case of weighted $3$-set packing, yields a $\max\{\frac{4}{3},α\}$-approximation for the MLSA in acyclic digraphs (dags), and provide a $\frac{7}{5}$-approximation for the hereditary $3$-set packing problem. In this paper, we obtain a $\frac{4}{3}$-approximation for the hereditary $3$-set packing problem, and, thus, also for the MLSA in dags. In doing so, we manage to leverage the full potential of the reduction provided by Fernandes and Lintzmayer. The algorithm that we study is a simple local search procedure considering swaps of size up to $10$. Its analysis relies on a two-stage charging argument.
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