Retraction: Improved Approximation Schemes for Dominating Set Problems in Unit Disk Graphs
Jittat Fakcharoenphol, Pattara Sukprasert
备注：
After posting the manuscript on arXiv, we were informed by Erik Jan van Leeuwen that both results were known and they appeared in his thesis[vL09]. We thank Erik Jan van Leeuwen for the information and we regret that we made this mistake
回撤注释：在arXiv上发布手稿后，Erik Jan van Leeuwen告诉我们，这两个结果都是已知的，并出现在他的论文中[vL 09]。MDS的PTAS见第79页的定理6.3.21，MCDS的PTAS见第82页的定理6.3.31。使用的技术非常相似。他指出，在移位技术中使用恒定数量的额外层来处理连接版本的想法不仅出现在Zhang et al. [ZGWD 09]但也在他2005年的论文[vL 05]中。最后，van Leeuwen还告诉我们，我们发布的开放问题已经被Marx解决了~[Mar 06，Mar 07]，他表明MDS的有效PTAS不存在[Mar 06]，并且在ETH下，$n^{O(1/ε)}$的运行时间是最好的[Mar 07]。我们感谢Erik Jan van Leeuwen提供的信息，我们很遗憾我们犯了这个错误。摘要撤回前：我们提出了两个（指数）更快的PTAS的支配集问题的单位盘图。给定一个单位圆盘图的几何表示，我们的PTAS的，找到$(1+ε)$ -近似解的最小支配集（MDS）和最小连通支配集（MCDS）的输入图运行在时间$n^{O(1/ε)}$。这可以与最著名的Nieberg和Hurink~[WAOA'05]针对MDS的仅使用图结构的$n^{O(1/ε
\log {1/ε})}$ -时间PTAS和Zhang、Gao、Wu和Du~[J Glob Optim'09]针对MCDS的$n^{O(1/ε^2)}$ -时间PTAS进行比较。我们的关键成分是改进的动态规划算法，指数依赖于更重要的一维“宽度”的问题。
英文摘要：
Retraction note: After posting the manuscript on arXiv, we were informed by Erik Jan van Leeuwen that both results were known and they appeared in his thesis[vL09]. A PTAS for MDS is at Theorem 6.3.21 on page 79 and A PTAS for MCDS is at Theorem 6.3.31 on page 82. The techniques used are very similar. He noted that the idea for dealing with the connected version using a constant number of extra layers in the shifting technique not only appeared Zhang et al.[ZGWD09] but also in his 2005 paper [vL05]. Finally, van Leeuwen also informed us that the open problem that we posted has been resolved by Marx~[Mar06, Mar07] who showed that an efficient PTAS for MDS does not exist [Mar06] and under ETH, the running time of $n^{O(1/ε)}$ is best possible [Mar07]. We thank Erik Jan van Leeuwen for the information and we regret that we made this mistake.
Abstract before retraction: 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 $(1+ε)$-approximate solutions to the Minimum Dominating Set (MDS) and the Minimum Connected Dominating Set (MCDS) of the input graph run in time $n^{O(1/ε)}$. This can be compared to the best known $n^{O(1/ε
\log {1/ε})}$-time PTAS by Nieberg and Hurink~[WAOA'05] for MDS that only uses graph structures and an $n^{O(1/ε^2)}$-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.
School of Statistics and Big Data, Zhengzhou College of Finance and Economics（郑州财经学院统计与大数据学院）；School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
Spanning Trees with a Small Vertex Cover: the Complexity on Specific Graph Classes
Graduate School of Information Sciences, Tohoku University（东北大学信息科学研究生院）；Center for Data-driven Science and Artificial Intelligence, Tohoku University（东北大学数据驱动科学与人工智能中心）
