In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to find $k$ pairwise vertex-disjoint paths such that the $i$-th path connects $s_i$ to $t_i$. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by $k$ unless coNP $\subseteq$ NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by $k + tw$, where $tw$ denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to $tw= k^{O(1)}$ under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time $2^{O(k^2)}\cdot n^{O(1)}$, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].

翻译：在平面不相交路径问题中，给定一个无向平面图及一组$k$个顶点对$(s_i,t_i)$，任务是找到$k$条两两顶点不相交的路径，使得第$i$条路径连接$s_i$与$t_i$。我们通过核化的视角研究该问题，旨在根据参数高效地缩减输入规模。我们证明平面不相交路径问题在以$k$为参数时不存在多项式核，除非coNP $\subseteq$ NP/poly，从而解决了[Bodlaender, Thomassé, Yeo, ESA'09]提出的开放问题。此外，我们排除了存在多项式图灵核的可能性，除非WK层级坍缩。我们的归约同样适用于边不相交路径的情形，而该情形即使在一般图中核化状态仍未明确。在积极方面，我们提出了一个以$k + tw$为参数的平面不相交路径问题的多项式核，其中$tw$表示输入图的树宽。结合我们的两个结果，我们排除了在上述相同假设下将树宽多项式时间（图灵）缩减至$tw = k^{O(1)}$的可能性。据我们所知，这是此类问题的首个困难性结果。最后，将我们的核与已知技术[Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94]相结合，得到了平面不相交路径问题可在$2^{O(k^2)}\cdot n^{O(1)}$时间内求解的另一（且更简化的）证明，与[Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20]的结果一致。