We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + ε)-approximation algorithm for this problem. We achieve this by utilizing the recent Lagrangian-multiplier preserving (LMP) primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al. (SWAT’16) and adopting an approach by Chudak et al. (Math. Prog. ’04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math. Prog. ’06). More generally, our approach readily gives a (4/3 ⋅ r + ε)-approximation on any graph class where the algorithm of Byrka et al. for the prize-collecting version gives an r-approximation. We argue that this can be interpreted as a generalization of an analogous result by Könemann et al. (Algorithmica ’11) for partial cover problems. Together with a lower bound construction by Mestre (STACS’08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al. in a non-black-box fashion could not beat the factor 4/3 ⋅ r when the tree merging step relies only on the solutions output by the LMP algorithm.

中文翻译：

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平面图上的节点加权k -MST逼近

我们研究在平面，节点加权图上找到最小权重的连接子图，该子图至少跨越k个顶点的问题。我们给一个（4 + ε）-这个问题的近似算法。我们通过利用Byrka等人的节点加权奖品收集Steiner树问题，利用最近的Lagrangian乘数保持（LMP）原始对偶3逼近来实现这一目标。（SWAT'16），并采用Chudak等人的方法。（Math。Prog。'04）关于边缘加权变体的拉格朗日松弛。尤其是，我们通过采用恒定数量的递归步骤并利用Arora和Karakostas的有限猜测程序（数学进展'06），改进了Sadeghian（2013）给出的其他顶点的拾取程序（树合并程序）。更一般地，我们的方法容易地给出了（4/3⋅ - [R + ε上的任何图形类） -近似其中Byrka等人的算法。奖品收集版本提供了一个r-近似 我们认为，这可以解释为Könemann等人对类似结果的概括。（Algorithmica '11）解决部分遮罩问题。连同梅斯特（Mestre）（STACS'08）的下界构造（用于部分掩盖），这意味着我们的界线在利用LMP算法作为拉格朗日弛豫的黑盒中的算法中基本上是最可能的。除此之外，我们还讨论了一个更复杂的下界构造，即使使用Byrka等人的LMP算法也是如此。在一个非黑盒时尚打不着因子4/3⋅ [R当树合并步骤只依靠由LMP算法输出的解决方案。