We present two (exponentially) faster PTAS's for dominating set problems in unit disk graphs. Given a geometric representation of a unit disk graph, our PTAS's that find $(1+\epsilon)$-approximate solutions to the Minimum Dominating Set (MDS) and the Minimum Connected Dominating Set (MCDS) of the input graph in time $n^{O(1/\epsilon)}$. This can be compared to the best known $n^{O(1/\epsilon \log {1/\epsilon})}$-time PTAS by Nieberg and Hurink [WAOA'05] for MDS that only uses graph structures and an $n^{O(1/\epsilon^2)}$-time PTAS for MCDS by Zhang, Gao, Wu, and Du [J Glob Optim'09]. Our key ingredients are improved dynamic programming algorithms that depend exponentially on more essential 1-dimensional "widths" of the problems.
翻译:我们用两个( expentially) 更快的 PTAS 表示单位磁盘图中设定的问题 。 鉴于单位磁盘图的几何表示, 我们的 PTAS 找到用于 MDS 最小支配数据集( MDS) 的$(1 ⁇ epsilon) 和 最小连接控制数据集( MDDS ) 。 这可以比对最著名的 $O( 1/\ epsilon\ log {1/\ epslon} 美元, 由 Nieberg 和 Hurink [WAOA' 05] 为 MDS 的 时间 PTAS, 它只使用张、 高、 吴 和 Du [J Glob Optim' 09] 的图形结构和$( $O1/\ epslon2) 。 我们的关键成份是指数化的动态编程算算法, 取决于问题的更基本的一维维的“ 维兹 ” 。