On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal

TL;DR

This paper investigates the polynomial kernelizability of the Odd Cycle Transversal problem under various structural parameterizations, providing positive results for some parameters and lower bounds for others, advancing understanding in parameterized complexity.

Contribution

It introduces a polynomial kernel for OCT when parameterized by the vertex deletion distance to bipartite graphs of bounded treewidth, and establishes kernelization lower bounds for other parameters.

Findings

Polynomial kernel for OCT with vertex deletion distance to bipartite graphs of bounded treewidth.

No polynomial kernel for OCT parameterized by distance to outerplanar graphs, assuming NP not in coNP/poly.

Lower bounds for kernelization with parameters like cluster, co-cluster graphs, and weighted OCT with vertex cover.

Abstract

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most l vertices. We study structural parameterizations of OCT with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Odd Cycle Transversal admits a polynomial kernel when parameterized by l. On the positive side, we show a polynomial kernel for OCT when parameterized by the vertex deletion distance to the class of bipartite graphs of treewidth at most w (for any constant w); this generalizes the parameter feedback vertex set number (i.e., the distance to a forest). Complementing this, we exclude polynomial kernels for OCT parameterized by the distance to outerplanar graphs, conditioned on the assumption that NP \not \subseteq coNP/poly. Thus the bipartiteness requirement for the treewidth w graphs is necessary. Further lower bounds are given for parameterization by distance from cluster and co-cluster graphs respectively, as well as for Weighted OCT parameterized by the vertex cover number (i.e., the distance from an independent set).