Kernel Bounds for Structural Parameterizations of Pathwidth

TL;DR

This paper investigates the kernelization complexity of the Pathwidth problem under various structural parameters, establishing lower bounds under certain conjectures and providing new polynomial kernels for specific parameters.

Contribution

It proves non-existence of polynomial kernels for Pathwidth under certain parameters assuming NP not in coNP/poly, and introduces new polynomial kernels for parameters like vertex cover and deletion distance to specific graph classes.

Findings

No polynomial kernel for Pathwidth when parameterized by vertex deletion distance to a clique, unless NP in coNP/poly.

Improved lower bounds for Treewidth kernelization based on cross-composition.

Polynomial kernels achieved for parameters such as vertex cover size and deletion distance to star or small components.

Abstract

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.