单位圆盘图最小团划分的弱鲁棒PTAS
A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
摘要：
研究了单位圆盘图的顶点划分问题。这个问题是NP困难的，已知有各种常数因子近似，目前最好的比率是3。我们的主要结果是对用边长表示的UDG的一个{ \em弱鲁棒}多项式时间近似方案（PTAS），它（i）计算团划分或（ii）证明图不是UDG：对于（i）计算团划分的情况，我们证明了如果输入是UDG，它保证在最优值的$(1+\eps)$比之内;然而，如果输入不是UDG，则它或者如在情况（i）中那样计算团分区而不保证团分区的质量，或者检测到它不是UDG。注意到，识别UDG的是NP难的，即使我们给定的边长，我们的PTAS是一个弱鲁棒算法。我们的算法可以转化为一个$O(\frac{\log^* n}{\eps^{O(1)}})$时间分布的PTAS。我们考虑一个加权版本的团划分问题的顶点加权UDG，推广的问题。我们注意到与未加权版本的一些关键区别，其中的想法有助于获得PTAS细分。然而，令人惊讶的是，它承认一个$(2+\eps)$ -近似算法的加权的情况下，图表示，说，作为一个邻接矩阵。这改进了以标准形式表示的UDG的{ \em未加权}情况的最佳已知的8-近似。
英文摘要：
We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a {\em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within $(1+\eps)$ ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG's is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(\frac{\log^* n}{\eps^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet, surprisingly, it admits a $(2+\eps)$-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the {\em unweighted} case for UDGs expressed in standard form.
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