Planar Disjoint Paths, Treewidth, and Kernels

TL;DR

This paper investigates the kernelization complexity of the Planar Disjoint Paths problem, proving it unlikely to admit polynomial kernels in general but providing a polynomial kernel when parameterized by k plus treewidth, and deriving implications for algorithmic runtime.

Contribution

It establishes the first hardness results for polynomial kernels in planar disjoint paths, and presents a polynomial kernel for the problem when parameterized by k plus treewidth.

Findings

No polynomial kernel exists for Planar Disjoint Paths unless coNP ⊆ NP/poly.

A polynomial kernel exists for Planar Disjoint Paths parameterized by k + treewidth.

Planar Disjoint Paths can be solved in time 2^{O(k^2)}·n^{O(1)}.

Abstract

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].