单位圆盘图中的半全控制
Semitotal domination in unit disk graphs
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中文总结 AI 辅助
针对单位圆盘图的最小半全控制问题，提出一种5因子近似算法，通过处理广度优先搜索树的层来构造满足条件的极大独立集，改进了之前运行时间为\(O(n^3)\)的5.75近似算法，新算法运行时间为\(O(n + m)\)（最坏\(O(n^2)\)）。
中文摘要 AI 辅助
集合\(S \subseteq V\)若满足：对于\(G=(V,E)\)，\(V \setminus S\)中的每个顶点都与\(S\)中的至少一个顶点相邻，且\(S\)中的每个顶点都与\(S\)中的另一个顶点距离在2以内，则称\(S\)是\(G\)的半全控制集。即使对于单位圆盘图，相应的决策问题也是NP完全的。本文针对基于图的输入模型中的单位圆盘图的最小半全控制问题，提出了一种5因子近似算法。该算法处理广度优先搜索树的各层，并构造一个满足半全条件的极大独立集。对于有\(n\)个顶点和\(m\)条边的图，算法运行时间为\(O(n + m)\)，最坏情况下为\(O(n^2)\)。这改进了之前已知的运行时间为\(O(n^3)\)的5.75近似算法。
英文摘要
A set $S \subseteq V$ is called a {\em semitotal dominating set} of $G=(V,E)$ if every vertex in $V \setminus S$ is adjacent to at least one vertex in $S$, and every vertex in $S$ is within distance 2 of another vertex in $S$. The corresponding decision problem is NP-complete even for unit disk graphs. In this paper, we present a 5-factor approximation algorithm for the Minimum Semitotal Domination problem on unit disk graphs in the graph-based input model. The algorithm processes the layers of a Breadth-First-Search tree and constructs a maximal independent set whose vertices satisfy the semitotal condition. For a graph with $n$ vertices and $m$ edges, the algorithm runs in $O(n + m)$ time, and hence in $O(n^2)$ time in the worst case. This improves the previously known 5.75-approximation algorithm with $O(n^3)$ running time.
