On Kernelization for Edge Dominating Set under Structural Parameters

TL;DR

This paper investigates the possibility of polynomial kernelizations for the Edge Dominating Set problem under various structural parameters, revealing both impossibility results and a complete classification for certain graph decompositions.

Contribution

It provides a full classification of when polynomial kernels exist for EDS parameterized by the deletion distance to graph classes, including surprising positive results for certain path-based parameters.

Findings

Polynomial kernels do not exist for deletion distance to disjoint unions of P3.

A polynomial kernel exists for deletion distance to disjoint unions of P5.

Complete classification for kernelizability based on graph class parameters.

Abstract

In the NP-hard Edge Dominating Set problem (EDS) we are given a graph $G=(V,E)$ and an integer $k$, and need to determine whether there is a set $F\subseteq E$ of at most $k$ edges that are incident with all (other) edges of $G$. It is known that this problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by $k$. A caveat for this parameter is that it needs to be large, i.e., at least equal to half the size of a maximum matching of $G$, for instances not to be trivially negative. Motivated by this, we study the existence of polynomial kernels for EDS when parameterized by structural parameters that may be much smaller than $k$. Unfortunately, at first glance this looks rather hopeless: Even when parameterized by the deletion distance to a disjoint union of paths $P_3$ of length two there is no polynomial kernelization (under standard assumptions), ruling out polynomial kernels for many smaller parameters like the feedback vertex set size. In contrast, somewhat surprisingly, there is a polynomial kernelization for deletion distance to a disjoint union of paths $P_5$ of length four. As our main result, we fully classify for all finite sets $\mathcal{H}$ of graphs, whether a kernel size polynomial in $|X|$ is possible when given $X$ such that each connected component of $G-X$ is isomorphic to a graph in $\mathcal{H}$.

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On Kernelization for Edge Dominating Set under Structural Parameters

Eva-Maria C. Hols

  

Stefan Kratsch

Unfortunately, at first glance this looks rather hopeless: Even when parameterized by the deletion distance to a disjoint union of paths $P_{3}$ of length two there is no polynomial kernelization (under standard assumptions), ruling out polynomial kernels for many smaller parameters like the feedback vertex set size. In contrast, somewhat surprisingly, there is a polynomial kernelization for deletion distance to a disjoint union of paths $P_{5}$ of length four. As our main result, we fully classify for all finite sets $\mathcal{H}$ of graphs, whether a kernel size polynomial in $|X|$ is possible when given $X$ such that each connected component of $G-X$ is isomorphic to a graph in $\mathcal{H}$.

1 Introduction

In the edge dominating set problem (EDS) we are given a graph $G=(V,E)$ and an integer $k$, and need to determine whether there is a set $F\subseteq E$ of at most $k$ edges that are incident with all (other) edges of $G$. It is known that this is equivalent to the existence of a maximal matching of size at most $k$. The edge dominating set problem is $\mathsf{NP}$-hard but admits a simple $2$-approximation by taking any maximal matching of $G$. It can be solved in time $\mathcal{O}^{*}(2.2351^{k})$111$\mathcal{O}^{*}$-notation hides factors that are polynomial in the input size. [IwaideN16], making it fixed-parameter tractable for parameter $k$. Additionally, for EDS any given instance $(G,k)$ can be efficiently reduced to an equivalent one $(G^{\prime},k^{\prime})$ with only $\mathcal{O}(k^{2})$ vertices and $\mathcal{O}(k^{3})$ edges [XiaoKP13] (this is called a kernelization).

The drawback of choosing the solution size $k$ as the parameter is that $k$ is large on many types of easy instances. This has been addressed for many other problems by turning to so called structural parameters that are independent of the solution size. Two lines of research in this direction have yielded polynomial kernels for several other $\mathsf{NP}$-hard problems. One possibility is to choose the parameter as the size of a set $X$ such that $G-X$ belongs to some class $\mathcal{C}$ where the problem in question can be efficiently solved; such sets $X$ are called modulators. The other possibility is to parameterize above some lower bound for the solution, i.e., the parameter is the difference between the solution size $k$ and the lower bound.

The vertex cover problem, where, given a graph $G$ and an integer $k$, we are asked whether there are $k$ vertices that are incident with all edges, has been successfully studied under different structural parameters. It had been observed that vertex cover is FPT parameterized by the size of a modulator to a class $\mathcal{C}$ when one can solve vertex cover on graphs that belong to $\mathcal{C}$ in polynomial time; e.g. if $\mathcal{C}$ is the graph class of forests or, more generally, of bipartite or Kőnig graphs. Furthermore, there also exist kernelizations for vertex cover parameterized by modulators to some graph classes $\mathcal{C}$. The first of a number of such results is due to Jansen and Bodlaender [JansenB13] who gave a kernelization with $\mathcal{O}(\ell^{3})$ vertices where $\ell$ is the size of a (minimum) feedback vertex set of the input graph. Clearly, the solution size $k$ cannot be bounded in terms of $\ell$ alone because forests already have arbitrarily large minimum vertex covers. This result has been generalized, e.g., for parameterization by the size of an odd cycle transversal [KratschW12].

There are also parameterized algorithms for vertex cover above lower bounds that address the specific complaint about the seemingly unnecessarily large parameter value $k$ in many graph classes. It was first shown that vertex cover parameterized by $\ell=k-MM$ where $MM$ stands for the size of a maximum matching is FPT [RamanRS11]. In other words, the parameter value $\ell$ is the difference between $k$ and the obvious lower bound. This has been improved to work also for parameterization by $\ell=k-LP$ where $LP$ stands for the minimum fractional vertex cover (as determined by the LP relaxation) [CyganPPW13, LokshtanovNRRS14] and, recently, even for parameter $\ell=k-(2LP-MM)$ [GargP16]. All of these above lower bound parameterizations of vertex cover also have randomized polynomial kernels [KratschW12, Kratsch16].

Motivated by the number of positive results for vertex cover parameterized by structural parameters we would like to know whether some of these results carry over to the related but somewhat more involved edge dominating set problem.

Our results.

For kernelization subject to the size of a modulator to some tractable class $\mathcal{C}$ there is bad news: Even if $\mathcal{C}$ contains only the disjoint unions of paths of length two (consisting of three vertices each) we show that there is no polynomial kernelization for parameterization by $|X|$ with $G-X\in\mathcal{C}$ unless $\mathsf{NP\subseteq coNP/poly}$ (and the polynomial hierarchy collapses). The same is true when $\mathcal{C}$ contains at least all disjoint unions of triangles. Thus, for the usual program of studying modulators to well-known hereditary graph classes $\mathcal{C}$ there is essentially nothing left to do because the only permissible connected components would have one or two vertices.222This very modest case actually admits a polynomial kernelization. That said, as the next result shows, this perspective would ignore an interesting landscape of positive and negative results that can be obtained by permitting certain forms of connected components in $G-X$ but not necessarily all induced subgraphs thereof, i.e., by dropping the requirement that $\mathcal{C}$ needs to be hereditary (closed under induced subgraphs).

Indeed, there is, e.g., a polynomial kernelization for parameter $|X|$ when all connected components of $G-X$ are paths of length four. This indicates that the structure even of constant-sized components permitted in $G-X$ determines in a nontrivial way whether or not there is a polynomial kernelization. Note the contrast with vertex cover where a modulator to component size $d$ admits a kernelization with $\mathcal{O}(k^{d})$ vertices for each fixed $d$. Naturally, we are interested in finding out exactly which cases admit polynomial kernels.

This brings us to our main result. For $\mathcal{H}$ a set of graphs, say that $G$ is an $\mathcal{H}$-component graph if each connected component of $G$ is isomorphic to some graph in $\mathcal{H}$. We fully classify the existence of polynomial kernels for parameterization by the size of a modulator to the class of $\mathcal{H}$-component graphs for all finite sets $\mathcal{H}$. To clarify, the input consists of $(G,k,X)$ such that $G-X$ is an $\mathcal{H}$-component graph and the task is to determine whether $G$ has an edge dominating set of size at most $k$; the parameter is $|X|$. Note that these problems are fixed-parameter tractable for all finite sets $\mathcal{H}$ because $G$ has treewidth at most $|X|+\mathcal{O}(1)$.

Theorem 1.1*.*

For every finite set $\mathcal{H}$ of graphs, the

To obtain the classification one needs to understand how connected components of $G-X$ that are isomorphic to some graph $H\in\mathcal{H}$ can interact with a solution for $G$, and to derive properties of $H$ that can be leveraged for kernels or lower bounds for kernelization. Crucially, edge dominating sets for $G$ may contain edges between $X$ and components of $G-X$. From the perspective of such a component (isomorphic to $H$) this is equivalent to first covering edges incident with some vertex set $B\subseteq V(H)$ (the endpoints of chosen edges to $X$) and then covering the remaining edges by a minimum edge dominating set for $H-B$. Depending on the size of a minimum edge dominating set of $H-B$ and further properties of $H$, such a set $B$ may be used to rule out any polynomial kernels or to give a lower bound of $\mathcal{O}(|X|^{d-\varepsilon})$ for the kernel size, where $d=|B|$. Conversely, absence of such sets or an upper bound for their size can be leveraged for kernels. Some sets $B$ may make others redundant, complicating both upper and lower bounds.

For a given finite set $\mathcal{H}$ of graphs, the lower bound obtained from the classification is simply the strongest one over all $H\in\mathcal{H}$. If this does not already rule out a polynomial kernelization then, for each $H\in\mathcal{H}$, we can reduce the number of components isomorphic to $H$ to $\mathcal{O}(|X|^{d(H)})$ where $d(H)$ depends only on $H$. Moreover, we also have the almost matching lower bound of $\mathcal{O}(|X|^{d(H)-\varepsilon})$, assuming $\mathsf{NP\nsubseteq coNP/poly}$. The value $d(\mathcal{H})$ is the maximum over all $d(H)$ for $H\in\mathcal{H}$ that yield such a polynomial lower bound; it can be computed in time depending only on $\mathcal{H}$, i.e., in constant time for each fixed $\mathcal{H}$.

Regarding parameterization above lower bounds, we prove that it is $\mathsf{NP}$-hard to determine whether a graph $G$ has an edge dominating set of size equal to the lower bound of half the size of a maximum matching. This rules out any positive results for parameter $\ell=k-\frac{1}{2}MM$.

Related work.

The parameterized complexity of

Problem 2*.*

Edge Dominating Set has been studied in a number of papers [Fernau06, FominGSS09, WangCFC09, Xiao10, XiaoKP13, XiaoN13a, EscoffierMPX15, IwaideN16]. Structural parameters were studied, e.g., by Escoffier et al. [EscoffierMPX15] who obtained an $\mathcal{O}^{*}(1.821^{\ell})$ time algorithm where $\ell$ is the vertex cover size of the input graph, and by Kobler and Rotics [KoblerR03] who gave a polynomial-time algorithm for graphs of bounded clique-width. It is easy to see that EDS is fixed-parameter tractable with respect to the treewidth of the input graph. Prieto [Prieto05] was the first to find a kernelization to $\mathcal{O}(k^{2})$ vertices for the standard parameterization by $k$; this was improved to $\mathcal{O}(k^{2})$ vertices and $\mathcal{O}(k^{3})$ edges by Xiao et al. [XiaoKP13] and further tweaked by Hagerup [Hagerup12]. Our work appears to be the first to study the existence of polynomial kernels for EDS subject to structural parameters, though some lower bounds, e.g., for parameter treewidth are obvious.

Classically,