Decremental Optimization of Dominating Sets Under the Reconfiguration Framework

TL;DR

This paper investigates the complexity of reducing dominating set sizes through elementary reconfiguration operations, establishing PSPACE-completeness in general but providing efficient algorithms for specific graph classes and parameterizations.

Contribution

It introduces the first complexity results for the decremental dominating set reconfiguration problem and offers fixed-parameter algorithms for certain graph parameters.

Findings

Problem is PSPACE-complete even for bipartite, split, and bounded pathwidth graphs.

Linear-time algorithms are provided for cographs, trees, and interval graphs.

The problem is W[2]-hard when parameterized by the size of the intermediary dominating set.

Abstract

Given a dominating set, how much smaller a dominating set can we find through elementary operations? Here, we proceed by iterative vertex addition and removal while maintaining the property that the set forms a dominating set of bounded size. This can be seen as the optimization variant of the dominating set reconfiguration problem, where two dominating sets are given and the question is merely whether they can be reached one from another through elementary operations. We show that this problem is PSPACE-complete, even if the input graph is a bipartite graph, a split graph, or has bounded pathwidth. On the positive side, we give linear-time algorithms for cographs, trees and interval graphs. We also study the parameterized complexity of this problem. More precisely, we show that the problem is W[2]-hard when parameterized by the upper bound on the size of an intermediary dominating set. On the other hand, we give fixed-parameter algorithms with respect to the minimum size of a vertex cover, or $d+s$ where $d$ is the degeneracy and $s$ is the upper bound of output solution.

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1 Introduction

Recently, Combinatorial reconfiguration [IDHPSUU11] has been extensively studied in the field of theoretical computer science. (See, e.g., surveys [H13, N18].) A reconfiguration problem is generally defined as follows: we are given two feasible solutions of a combinatorial search problem, and asked to determine whether we can transform one into the other via feasible solutions so that all intermediate solutions are obtained from the previous one by applying the specified reconfiguration rule. This framework is applied to several well-studied combinatorial search problems; for example, Independent Set [BB17, HD05, KMM12, LM18], Vertex Cover [MNR14, MNRSS17], Dominating Set [HIMNOST16, LMPRS18, MNRSS17, SMN16], and so on.

The Dominating Set Reconfiguration problem is one of the well-studied reconfiguration problems. For a graph $G=(V,E)$, a vertex subset $D\subseteq V$ is called a dominating set of $G$ if $D$ contains at least one vertex in the closed neighborhood of each vertex in $V$. Figure 1 illustrates four dominating sets of the same graph. Suppose that we are given two dominating sets $D_{\sf 0}$ and $D_{\sf t}$ of a graph whose cardinalities are at most a given upper bound $k$. Then the Dominating Set Reconfiguration problem asks to determine whether we can transform $D_{\sf 0}$ into $D_{\sf t}$ via dominating sets of cardinalities at most $k$ such that all intermediate ones are obtained from the previous one by adding or removing exactly one vertex. Note that this reconfiguration rule, i.e. adding or removing exactly one vertex while keeping the cardinality constraint, is called the token addition and removal (TAR) rule. Figure 1 illustrates an example of transformation between two dominating sets $D_{0}$ and $D_{3}$ for an upper bound $k=4$.

Combinatorial reconfiguration models “dynamic” transformations of systems, where we wish to transform the current configuration of a system into a more desirable one by a step-by-step transformation. In the current framework of combinatorial reconfiguration, we need to have in advance a target (a more desirable) configuration. However, it is sometimes hard to decide a target configuration, because there may exist exponentially many desirable configurations. Based on this situation, Ito et al. introduced the new framework of reconfiguration problems, called optimization variant [OPT-ISR]. In this variant, we are given a single solution as a current configuration, and asked for a more “desirable” solution reachable from the given one. This variant was introduced very recently, hence it has only been applied to Independent Set Reconfiguration to the best of our knowledge. Therefore and since Dominating Set Reconfiguration is one of the well-studied reconfiguration problems as we already said, we focus on this problem and study it under this framework.

1.1 Our problem

In this paper, we study the optimization variant of Dominating Set Reconfiguration, denoted by OPT-DSR. To avoid confusion, we call the original Dominating Set Reconfiguration the reachability variant, and we denote it by REACH-DSR. Suppose that we are given a graph $G$, two integers $k,s$, and a dominating set $D$ of $G$ whose cardinality is at most $k$; we call $k$ an upper bound and $s$ a solution size. Then OPT-DSR asks for a dominating set $D_{\sf t}$ satisfying the following two conditions: (a) the cardinality of $D_{\sf t}$ is at most $s$, and (b) $D_{\sf t}$ can be transformed from $D$ under the TAR rule with upper bound $k$. For example, if we are given a dominating set $D_{0}$ in Figure 1 and two integers $k=4$ and $s=2$, then one of the solutions is $D_{3}$, because $D_{3}$ can be transformed from $D_{0}$ and $|D_{3}|\leq 2$ holds.

1.2 Related results

Although OPT-DSR is being introduced in this paper, some results for REACH-DSR relate to OPT-DSR in the sense that the techniques to show the computational hardness or construct an algorithm will be used in our proof for OPT-DSR. We thus list such results for REACH-DSR in the following.

There are several results for the polynomial-time solvability of REACH-DSR. Haddadan et al. [HIMNOST16] showed that REACH-DSR under TAR rule is PSPACE-complete for split graphs, for bipartite graphs, and for planar graphs, while linear-time solvable for interval graphs, for cographs, and for forests. REACH-DSR is also studied well from the viewpoint of fixed-parameter (in)tractability. Mouawad et al. [MNRSS17] showed that REACH-DSR under TAR is W[2]-hard when parameterized by an upper bound $k$. As a positive result, Lokshtanov et al. [LMPRS18] gave a fixed-parameter algorithm with respect to $k+d$ for graphs that exclude $K_{d,d}$ as a subgraph.

Ito et al. studied optimization variant of Independent Set Reconfiguration (OPT-ISR) [OPT-ISR]. More precisely, they proved that this problem is PSPACE-hard on bounded pathwidth, NP-hard on planar graphs, while linear-time solvable on chordal graphs. They also gave an XP-algorithm with respect to the solution size, and a fixed-parameter algorithm with respect to both solution size and degeneracy.

1.3 Our results

In this paper, we study OPT-DSR from the viewpoint of the polynomial-time (in)tractability and fixed-parameter (in)tractability.

We first study the polynomial-time solvability of OPT-DSR with respect to graph classes (See Figure 2). Specifically, we show that the problem is PSPACE-complete even for split graphs, for bipartite graphs, and for bounded pathwidth graphs, and NP-hard for planar graphs with bounded maximum degree. On the other hand, the problem is linear-time solvable for cographs, trees and interval graphs. The inclusions of these graph classes are represented in Figure 2.

We then study the fixed-parameter (in)tractability of OPT-DSR. We first focus on the following four graph parameters: the degeneracy $d$, the maximum degree $\Delta$, the pathwidth $pw$, and the vertex cover number $\tau$ (that is the size of a minimum vertex cover). Figure 3(a) illustrates the relationship between these parameters, where A $\rightarrow$ B means that the parameter A is bounded by some function of B. This relation implies that if we have a result stating that OPT-DSR is fixed-parameter tractable for A then the tractability for B follows, while if we have a negative (i.e. intractability) result for B then it extends to A. From results for polynomial-time solvability, we show the PSPACE-completeness for fixed $pw$ and NP-hardness for fixed $\Delta$, and hence the problem is fixed-parameter intractable for each parameter $pw$, $\Delta$ and $d$ under P $\neq$ PSPACE or P $\neq$ NP. As a positive result, we give an FPT algorithm for $\tau$.We then consider two input parameters: the solution size $s$ and the upper bound $k$. (See Figure 3(c).) We show that OPT-DSR is W[2]-hard when parameterized by $k$. We note that we can assume without loss of generality that $s<k$ holds, as explained in Section 2. Therefore, it immediately implies W[2]-hardness for $s$. Most single parameters (except for $\tau$) cause a negative (intractability) result. We thus finally consider combinations of one graph parameter and one input parameter. We give an FPT algorithm with respect to $s+d$. (See Figure 3(b).) In the end, we can conclude from the discussion above that for any combination of a graph parameter $p\in\{d,\Delta,pw,\tau\}$ and an input parameter $q\in\{s,k\}$, OPT-DSR is fixed-parameter tractable when parameterized by $p+q$.

2 Preliminaries

For a graph $G$, we denote by $V(G)$ and $E(G)$ the vertex set of $G$ and edge set of $G$, respectively. For a vertex $v\in V(G)$, we let $N_{G}(v)=\{w\mid vw\in E(G)\}$ and $N_{G}[v]=N_{G}(v)\cup\{v\}$; we call a vertex in $N_{G}(v)$ a neighbor of $v$ in $G$. For a vertex subset $S\subseteq V(G)$, we let $N_{G}[S]=\bigcup_{v\in S}N_{G}[v]$. If there is no confusion, we sometimes omit $G$ from the notation.

2.1 Optimization variant of Dominating Set Reconfiguration

For a graph $G=(V,E)$, a vertex subset $D\subseteq V$ is a dominating set of $G$ if $N[D]=V(G)$. For a dominating set $D$, we say that $u\in D$ dominates $v\in V$ if $v\in N[u]$ holds. We say that a vertex $v\in D$ has a private neighbor in $D$ if there exists a vertex $u\in N[v]$ such that $N[u]\cap D=\{v\}$. In other words, the vertex $u$ is dominated only by $v$ in $D$. Note that the private neighbor of a vertex can be itself. A dominating set is (inclusion-wise) minimal if and only if each of its vertices has a private neighbor, and minimum if and only if the cardinality is minimum among all dominating sets. Notice that any minimum dominating set is minimal.

Let $D$ and $D^{\prime}$ be two dominating sets of $G$. We say that $D$ and $D^{\prime}$ are adjacent if $|D\Delta D^{\prime}|=1$, where $D\Delta D^{\prime}=(D\setminus D^{\prime})\cup(D^{\prime}\setminus D)$ and we denote this by $D\leftrightarrow D^{\prime}$. Let us now assume that $D$ and $D^{\prime}$ are both of size at most $k$, for some given $k\geq 0$. Then, a reconfiguration sequence between $D$ and $D^{\prime}$ under the TAR rule (or sometimes called a TAR-sequence) is a sequence $\langle D=D_{0},D_{1},\ldots,D_{\ell}=D^{\prime}\rangle$ of dominating sets of $G$ such that:

• •

for each $i\in\{0,1,\ldots,\ell\}$, $D_{i}$ is a dominating set of $G$ such that $|D_{i}|\leq k$; and

• •

for each $i\in\{0,1,\ldots,\ell-1\}$, $D_{i}\leftrightarrow D_{i+1}$ holds.

Considering a reconfiguration sequence under the TAR rule, we sometimes write TAR($k$) instead of TAR to emphasize the upper bound $k$ on the size of a solution. We say that $D^{\prime}$ is reachable from $D$ if there exists a reconfiguration sequence between $D$ and $D^{\prime}$; since a reconfiguration sequence is reversible, if $D^{\prime}$ is reachable from $D$, then $D$ is also reachable from $D^{\prime}$. We write $D\overset{k}{\leftrightsquigarrow}D^{\prime}$ if $D^{\prime}$ (resp. $D$) is reachable from $D$ (resp. $D^{\prime}$). Then, the optimization variant of the Dominating Set Reconfiguration problem (OPT-DSR) is defined as follows: