Planar Disjoint Paths, Treewidth, and Kernels
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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 ⊆ NP/poly, resolving an open problem by [Bodlaender, Thomassé, 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)· n^O(1), matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].
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