Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

TL;DR

This paper investigates the kernelization complexity of a broad class of graph recognition problems, revealing that polynomial kernels exist only under specific structural conditions related to forbidden subgraphs.

Contribution

It establishes a precise criterion for when fixed-parameter recognition problems admit polynomial kernels, based on the presence of small forbidden induced subgraphs, and employs pushing process techniques.

Findings

Polynomial kernels exist if and only if the forbidden set contains a graph with at most 2 vertices.

The study links kernelization feasibility to structural properties of forbidden subgraphs.

Uses pushing process technique for both kernelization and lower bound proofs.

Abstract

A fundamental graph problem is to recognize whether the vertex set of a graph $G$ can be bipartitioned into sets $A$ and $B$ such that $G[A]$ and $G[B]$ satisfy properties $\Pi_A$ and $\Pi_B$, respectively. This so-called $(\Pi_A,\Pi_B)$-Recognition problem generalizes amongst others the recognition of $3$-colorable, bipartite, split, and monopolar graphs. In this paper, we study whether certain fixed-parameter tractable $(\Pi_A,\Pi_B)$-Recognition problems admit polynomial kernels. In our study, we focus on the first level above triviality, where $\Pi_A$ is the set of $P_3$-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph $G[A]$, and $\Pi_B$ is characterized by a set $\mathcal{H}$ of connected forbidden induced subgraphs. We prove that, under the assumption that NP is not a subset of coNP/poly, \textsc{$(\Pi_A,\Pi_B)$-Recognition} admits a polynomial kernel if and only if $\mathcal{H}$ contains a graph with at most $2$ vertices. In both the kernelization and the lower bound results, we exploit the properties of a pushing process, which is an algorithmic technique used recently by Heggerness et al. and by Kanj et al. to obtain fixed-parameter algorithms for many cases of $(\Pi_A,\Pi_B)$-Recognition, as well as several other problems.