Introducing lop-kernels: a framework for kernelization lower bounds

TL;DR

This paper introduces lop-kernels, a framework for establishing kernelization lower bounds in optimization problems, demonstrating the near-optimality of existing kernels for MMVC and other problems, and providing new subquadratic kernels for specific graph classes.

Contribution

The paper presents a general framework for kernelization lower bounds called lop-kernels, and applies it to show the optimality of existing kernels and develop new kernels for MMVC and related problems.

Findings

Quadratic kernel for MMVC is essentially optimal.

Cubic kernel for Feedback Vertex Set is essentially optimal.

Subquadratic kernels for MMVC on H-free graphs using Erdős-Hajnal property.

Abstract

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph $G$ and a positive integer $k$, and the objective is to decide whether $G$ contains a minimal vertex cover of size at least $k$. Motivated by the kernelization of MMVC with parameter $k$, our main contribution is to introduce a simple general framework to obtain kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop-kernels. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on $K_t$-free graphs. Back to the MMVC problem, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on $H$-free graphs for several graphs $H$, such as the bull, the paw, or the complete graphs, by making use of the Erd\"os-Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$.