For all $k\ge 1$, we show that deciding whether a graph is k-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is NP-complete. In particular, given a graph with local crossing number either at most $k\ge 1$ or at least 2k, we show that it is NP-complete to decide whether the local crossing number is at most k or at least 2k. This algorithmic lower bound proves the non-existence of a $(2-ϵ)$-approximation algorithm for any fixed $k\ge 1$. In addition, we analyze the sometimes competing relationship between the local crossing number (maximum number of crossings per edge) and crossing number (total number of crossings) of a drawing. We present results regarding the non-existence of drawings that simultaneously approximately minimize both the local crossing number and crossing number of a graph.

对全部 $ķ\ge \mathrm{1个}$，我们证明了确定图是否为k平面是NP完全的，从而扩展了确定1平面为NP完全的众所周知的事实。此外，我们证明了该决策问题的缺口版本是NP完全的。特别地，给定一个最多具有局部交叉数的图$ķ\ge \mathrm{1个}$或至少2 k，我们证明判定本地交叉数最多是k还是至少2 k是NP完全的。该算法的下界证明了a的不存在$（2-ϵ）$-固定的近似算法 $ķ\ge \mathrm{1个}$。此外，我们分析了图形的局部交叉数（每个边的最大交叉数）和交叉数（交叉的总数）之间有时存在的竞争关系。我们提出了与不存在的图形有关的结果，这些结果同时使图形的局部交叉数和交叉数同时最小化。