[Paper Review] Smaller parameters for vertex cover kernelization

Eva-Maria C. Hols, Stefan Kratsch|arXiv (Cornell University)|Nov 13, 2017

Advanced Graph Theory Research16 references3 citations

TL;DR

This paper presents polynomial kernelization for the vertex cover problem using structural parameters based on modulators to d-quasi-forests, d-quasi-bipartite graphs, and d-quasi-integral graphs. It establishes tight bounds on minimal blocking sets (size ≤ d+2 for d-quasi-forests and d-quasi-bipartite graphs, and ≤ 2d+2 for d-quasi-integral graphs), leading to kernel sizes of O(|X|^{3d+9}) and O(|X|^{2d+3}) respectively, with matching lower bounds, generalizing prior results and unifying multiple kernelization frameworks.

ABSTRACT

We revisit the topic of polynomial kernels for Vertex Cover relative to structural parameters. Our starting point is a recent paper due to Fomin and Strømme [WG 2016] who gave a kernel with $\mathcal{O}(|X|^{12})$ vertices when $X$ is a vertex set such that each connected component of $G-X$ contains at most one cycle, i.e., $X$ is a modulator to a pseudoforest. We strongly generalize this result by using modulators to $d$-quasi-forests, i.e., graphs where each connected component has a feedback vertex set of size at most $d$, and obtain kernels with $\mathcal{O}(|X|^{3d+9})$ vertices. Our result relies on proving that minimal blocking sets in a $d$-quasi-forest have size at most $d+2$. This bound is tight and there is a related lower bound of $\mathcal{O}(|X|^{d+2-ε})$ on the bit size of kernels. In fact, we also get bounds for minimal blocking sets of more general graph classes: For $d$-quasi-bipartite graphs, where each connected component can be made bipartite by deleting at most $d$ vertices, we get the same tight bound of $d+2$ vertices. For graphs whose connected components each have a vertex cover of cost at most $d$ more than the best fractional vertex cover, which we call $d$-quasi-integral, we show that minimal blocking sets have size at most $2d+2$, which is also tight. Combined with existing randomized polynomial kernelizations this leads to randomized polynomial kernelizations for modulators to $d$-quasi-bipartite and $d$-quasi-integral graphs. There are lower bounds of $\mathcal{O}(|X|^{d+2-ε})$ and $\mathcal{O}(|X|^{2d+2-ε})$ for the bit size of such kernels.

Motivation & Objective

• To generalize existing polynomial kernelizations for vertex cover beyond modulators to forests and pseudoforests by introducing d-quasi-forests, d-quasi-bipartite, and d-quasi-integral graphs as structural parameters.
• To establish tight upper bounds on the size of minimal blocking sets in these graph classes, which are crucial for kernel size guarantees.
• To unify and extend prior kernelization results for vertex cover parameterized by feedback vertex set, degree-bounded modulators, and parameterization above the fractional vertex cover optimum.
• To prove matching lower bounds for kernel size, showing that the derived kernel sizes are asymptotically optimal under standard complexity assumptions.

Proposed method

• Define d-quasi-forests as graphs where each connected component has a feedback vertex set of size at most d, and similarly define d-quasi-bipartite and d-quasi-integral graphs.
• Prove that minimal blocking sets in d-quasi-forests and d-quasi-bipartite graphs have size at most d+2, and in d-quasi-integral graphs at most 2d+2, using structural graph properties and bounds on independent sets.
• Use reduction rules to transform the input graph into an equivalent instance where G−X has at most |X|^{2d+3} connected components, leveraging polynomial-time computation of maximum independent sets in d-quasi-integral graphs.
• Apply existing randomized polynomial kernelization for vertex cover above the LP relaxation to the transformed instance, ensuring the kernel size depends polynomially on |X|.
• Establish that the kernel size is O(|X|^{3d+9}) for d-quasi-forests and O(|X|^{2d+3}) for d-quasi-integral graphs, based on the bounded size of minimal blocking sets.
• Derive matching lower bounds of O(|X|^{d+2−ε}) and O(|X|^{2d+2−ε}) for the bit size of kernels, assuming NP ⊈ coNP/poly.

Experimental results

Research questions

• RQ1 Can polynomial kernelization for vertex cover be extended beyond modulators to forests and pseudoforests to more general graph classes?
• RQ2 What is the maximum size of a minimal blocking set in a d-quasi-forest, and how does this affect kernel size?
• RQ3 Can the framework of modulators to d-quasi-forests be generalized to d-quasi-bipartite and d-quasi-integral graphs while maintaining polynomial kernel size?
• RQ4 Are the derived kernel sizes asymptotically optimal, and what are the lower bounds for kernel size in terms of the parameter |X|?
• RQ5 Can existing kernelization techniques for parameterization above the fractional vertex cover be unified under a single framework using these new graph classes?

Key findings

• The size of minimal blocking sets in d-quasi-forests and d-quasi-bipartite graphs is at most d+2, and this bound is tight, as shown by the existence of cliques of size d+2 in these classes.
• The size of minimal blocking sets in d-quasi-integral graphs is at most 2d+2, and this bound is also tight, with cliques of size 2d+2 serving as matching lower-bound constructions.
• A polynomial kernel of size O(|X|^{3d+9}) is achieved for vertex cover parameterized by a modulator X to a d-quasi-forest, generalizing previous results for modulators to forests and pseudoforests.
• For modulators to d-quasi-integral graphs, a kernel of size O(|X|^{2d+3}) is obtained, which subsumes kernelizations parameterized by feedback vertex set, maximum degree at most two, and parameterization above the fractional vertex cover.
• Matching lower bounds of O(|X|^{d+2−ε}) and O(|X|^{2d+2−ε}) are proven for the bit size of kernels, assuming NP ⊈ coNP/poly, showing that the kernel sizes are asymptotically optimal.
• The results unify and extend multiple existing kernelization frameworks, including those for modulators to forests, degree-bounded graphs, and parameterization above LP relaxation.