Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

TL;DR

This paper investigates the kernelization of Vertex Cover through the lens of elimination distances and blocking sets, establishing bounds and conditions for polynomial kernels across various graph classes.

Contribution

It unifies and generalizes kernelization results for Vertex Cover using blocking sets and elimination distances, providing tight bounds and new polynomial kernelizations.

Findings

Bounded minimal blocking set size is necessary but not sufficient for polynomial kernels.

Exact maximum size of minimal blocking sets is determined for graphs with bounded elimination distance.

Polynomial kernels are derived for Vertex Cover parameterized by deletion sets to certain graph classes.

Abstract

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class $\mathcal{C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class $\mathcal{C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes $\mathcal{C}$, including the class $\mathcal{C}_{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or $\mathcal{C}_{LP}$ graphs.

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Full text

We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class $\mathcal{C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components.

We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class $\mathcal{C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes $\mathcal{C}$, including the class $\mathcal{C}_{\mathrm{LP}}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or $\mathcal{C}_{\mathrm{LP}}$ graphs.

1 Introduction

In the Vertex Cover problem we are given an undirected graph $G=(V,E)$ and an integer $k$; the question is whether there exists a set $S\subseteq V$ of at most $k$ vertices such that each edge of $G$ is incident with a vertex of $S$, or, in other words, such that $G-S$ is an independent set. Despite Vertex Cover being $\mathsf{NP}$-complete, it is known that there are efficient preprocessing algorithms that reduce any instance $(G,k)$ to an equivalent instance with $\mathcal{O}(k^{2})$ or even at most $2k$ vertices (and size polynomial in $k$). The existence or non-existence of such preprocessing routines for $\mathsf{NP}$-hard problems has been studied intensively in the field of parameterized complexity under the term polynomial kernelization,111A polynomial kernelization is an efficient algorithm that given any instance with parameter value $\ell$ returns an equivalent instance of size polynomial in $\ell$. In the initial Vertex Cover example we have $\ell=k$ but many other parameters will be considered. Full definitions can be found in Section LABEL:section::preliminaries. and Vertex Cover has turned out to be one of the most fruitful research subjects with a variety of upper and (conditional) lower bounds subject to different parameters (see, e.g., [BougeretS17]).

In the present work, we seek to unify and generalize existing results by using so-called blocking sets. A blocking set in a graph $G=(V,E)$ is any set $Y\subseteq V$ that is not a subset of any minimum cardinality vertex cover of $G$, e.g., the set $V$ itself is always a blocking set. Of particular interest are minimal blocking sets, i.e., those that are minimal under set inclusion. Several graph classes have constant upper bounds on the size of minimal blocking sets, e.g., in any forest (or even in any bipartite graph) every minimal blocking set has size at most two. On the other hand, even restrictive classes like outerplanar graphs have unbounded minimal blocking set size, i.e., for each $d$ there is a graph in the class with a minimal blocking set of size greater than $d$. As a final example, cliques are the unique graphs for which $V$ is the only (minimal) blocking set because all optimal vertex covers have form $V\setminus\{v\}$ for any $v\in V$; in particular, any graph class containing all cliques has unbounded minimal blocking set size.

For ease of reading, the introduction focuses mostly on hereditary graph classes, i.e., those closed under vertex deletion; the later sections give more general results modulo suitable technical conditions, where appropriate. Apart from that, throughout the paper, we restrict study to graph classes $\mathcal{C}$ that are robust, that is, they are closed under disjoint union and under deletion of connected components. In other words, a graph $G$ is in $\mathcal{C}$ if and only if all of its connected components belong to $\mathcal{C}$. Most graph classes studied in the context of kernels for Vertex Cover are robust, and a large number of them are also hereditary. A particular non-hereditary graph class of interest for us is the class $\mathcal{C}_{\mathrm{LP}}$ of graphs whose minimum vertex cover size equals the minimum size of a fractional vertex cover (denoted $\mathrm{LP}$ as it is also the optimum solution value for the vertex cover LP relaxation).

1.1 Blocking sets and kernels for Vertex Cover

Most known polynomial kernelizations for Vertex Cover are for parameterization by the vertex deletion distance to some fixed hereditary graph class $\mathcal{C}$ that is also robust, e.g., for $\mathcal{C}$ being the class of forests [JansenB13], graphs of maximum degree one or two [MajumdarRS18], pseudoforests [FominS16] (each component has at most one cycle), bipartite graphs [KratschW12], $d$-quasi forests/bipartite graphs [HolsK17] (at most $d$ vertex deletions per component away from being a forest/bipartite), cluster graphs of bounded clique size [MajumdarRS18], or graphs of bounded treedepth [BougeretS17]. Concretely, the input is of form $(G,k,X)$, asking whether $G$ has a vertex cover of size at most $k$, where $X\subseteq V$ such that $G-X\in\mathcal{C}$; the size $\ell=|X|$ of the modulator $X$ is the parameter. Blocking sets have been implicitly or explicitly used for most of these results and we point out that all the mentioned classes have bounded minimal blocking set size.

As our first result, we show that this is not a coincidence: If $\mathcal{C}$ is closed under disjoint union (or, more strongly, if $\mathcal{C}$ is robust) then bounded size of minimal blocking sets in graphs of $\mathcal{C}$ is necessary for a polynomial kernel to exist (Section LABEL:section::kernelimpliesbmbss). Moreover, the maximum size of minimal blocking sets in $\mathcal{C}$ yields a lower bound for the possible kernel size.

Theorem 1.1*.*

Let $\mathcal{C}$ be a graph class that is closed under disjoint union. If $\mathcal{C}$ contains any graph with a minimal blocking set of size $d$ then

To the best of our knowledge, this theorem captures all known kernel lower bounds for

Problem 2*.*

Vertex Cover parameterized by deletion distance to any union-closed graph class $\mathcal{C}$, e.g., ruling out polynomial kernels for $\mathcal{C}$ being the class of mock forests (each vertex is in at most one cycle) [FominS16], outerplanar graphs [Jansen13Thesis], or any class containing all cliques [BodlaenderJK14]; and getting kernel size lower bounds for graphs of bounded treedepth [BougeretS17] or cluster graphs of bounded clique size [MajumdarRS18]. To get lower bounds of this type, it now suffices to prove (or observe) that $\mathcal{C}$ has large or even unbounded minimal blocking set size.

It is natural to ask whether the converse holds, i.e., whether a bound on the minimal blocking set size directly implies the existence of a polynomial kernelization. Unfortunately, we show that this does not hold in a strong sense: There is a class $\mathcal{C}$ such that all graphs in $\mathcal{C}$ have minimal blocking sets of size one, but there is no polynomial kernelization (Section LABEL:section::bmbssisnotsufficient). More strongly, solving