(Total) Vector Domination for Graphs
with Bounded Branchwidth

The dominating set problem itself is one of the most fundamental graph optimization problems, and it has been intensively and extensively studied from many points of view. In the sense that the vector or multiple domination contains the setting of not only the ordinary dominating set problem but also many variants, there are an enormous number of related studies. Here we pick some representatives up.

As a research of the domination problems from the viewpoint of the algorithm design, Cicalese, Milanic and Vaccaro gave detailed analyses of the approximability and inapproximability [8, 9]. They also provided some exact polynomial-time algorithms for special classes of graphs, such as complete graphs, trees, $P_{4}$-free graphs, and threshold graphs.

For graphs with bounded treewidth (or branchwidth), the ordinary domination problems can be solved in polynomial time. As for the fixed-parameter tractability, it is known that even the ordinary dominating set problem is W[2]-complete with respect to solution size $k$; it is unlikely to be fixed-parameter tractable [16]. In contrast, it can be solved in $O(2^{15.13\sqrt{k}}+n^{3})$ time for planar graphs, that is, it is subexponential fixed-parameter tractable [18]. The subexponent part comes from the inequality $bw(G)\leq 12\sqrt{k}+9$, where $k$ is the size of a dominating set of $G$. Behind the inequality, there is a unified property of parameters, called bidimensionality [14]. Namely, the subexponential fixed-parameter algorithm of the dominating set for planar graphs (more precisely, $H$-minor-free graphs [13]) is based on the bidimensionality.

A maximization version of the ordinary dominating set is also considered. Partial Dominating Set is the problem of maximizing the number of vertices to be dominated by using a given number $k$ of vertices. In [2], it was shown that partial dominating set problem is FPT with respect to $k$ for $H$-minor-free graphs. Later, [17] gives a subexponential FPT with respect to $k$ for apex-minor-free graphs, also a super class of planar graphs. Although partial dominating set is an example of problems to which the bidimensionality theory cannot be applied, they develop a technique to reduce an input graph so that its treewidth becomes $O(\sqrt{k})$.

For the vector domination, a polynomial-time algorithm for graphs of bounded treewidth has been proposed very recently [7], as mentioned before. In [25], it is shown that the vector domination for $\rho$-degenerated graphs can be solved in $k^{O(\rho k^{2})}n^{O(1)}$ time, if $d(v)>0$ holds for $\forall v\in V$ (positive constraint). Since any planar graph is $5$-degenerated, the vector domination for planar graphs is fixed-parameter tractable with respect to solution size, under the positive constraint. Furthermore, the case where $d(v)$ could be $0$ for some $v$ can be easily reduced to the positive case by using the transformation discussed in [3], with increasing the degeneracy only $1$. It follows that the vector domination for planar graphs is FPT with respect to solution size $k$. However, for the total vector domination and multiple domination, neither polynomial time algorithm for graphs of bounded treewidth nor fixed-parameter algorithm for planar graphs has been known.

Other than these, several generalized versions of the dominating set problem are also studied. $(k,r)$-center problem is the one that asks the existence of set $S$ of $k$ vertices satisfying that for every vertex $v\in V$ there exists a vertex $u\in S$ such that the distance between $u$ and $v$ is at most $r$; $(k,1)$-center corresponds to the ordinary dominating set. The $(k,r)$-center for planar graphs is shown to be fixed-parameter tractable with respect to $k$ and $r$ [12]. For $\sigma,\rho\subseteq\{0,1,2,\ldots\}$ and a positive integer $k$, $\exists[\sigma,\rho]$-dominating set is the problem that asks the existence of set $S$ of $k$ vertices satisfying that $|N(v)\cap S|\in\sigma$ holds for $\forall v\in S$ and $|N(v)\cap S|\in\rho$ for $\forall v\in V\setminus S$, where $N(v)$ denotes the open neighborhood of $v$. If $\sigma=\{0,1,\ldots\}$ and $\rho=\{1,2,\ldots\}$, $\exists[\sigma,\rho]$-dominating set is the ordinary dominating set problem, and if $\sigma=\{0\}$ and $\rho=\{0,1,2,\ldots\}$, it is the independent set. In [6], the parameterized complexity of $\exists[\sigma,\rho]$-dominating set with respect to treewidth is also considered.

Our results are summarized as follows:

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  We present a polynomial-time algorithm for the vector domination of graph $G=(V,E)$ with bounded branchwidth. The running time is roughly $O(n^{6bw(G)+2})$.

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  We present polynomial-time algorithms for the total vector domination and multiple domination of graph $G$ with bounded branchwidth. The running time is roughly $O(2^{9bw(G)/2}$ $n^{6bw(G)+2})$.

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  Let $G$ be a planar graph. Then, we can check in $O(n^{4}+\min\{k,d^{*}\}^{40\sqrt{k}+34}n)$ time whether $G$ has a vector dominating set with cardinality at most $k$ or not, where $d^{*}=\max\{d(v)\mid v\in V\}$.

• •

  Let $G$ be a planar graph. Then, we can check in $O(n^{4}+2^{30\sqrt{k}+51/2}\min\{k,d^{*}\}^{40\sqrt{k}+34}n)$ time whether $G$ has a total vector dominating set and a multiple dominating set with cardinality at most $k$ or not.

It should be noted that it is actually possible to design directly polynomial time algorithms for graphs with bounded treewidth, but they are slower than the ones for graphs with bounded branchwidth; this is the reason why we adopt the branchwidth instead of the treewidth.

As far as the authors know, the second and fourth results give the first polynomial time algorithms and the first fixed-parameter algorithm for the total vector domination and multiple domination of graphs with bounded branchwidth (or treewidth) and planar graphs, respectively. As for the vector domination, we give an $O(n^{6bw(G)+2})$-time algorithm, whose running time is $O(n^{6(tw(G)+1)+2})$ in terms of the treewidth, whereas the recent paper [7] gives an $O(cw(G)|\sigma|(n+1)^{5cw(G)})$-time algorithm, where $|\sigma|$ is the encoding length of $k$-expression used in the algorithm, and is bounded by a polynomial in the input size for fixed $k$. Since $cw(G)\leq 2^{tw(G)+1}+1$ holds, this is an $O(2^{tw(G)+1}|\sigma|(n+1)^{5\cdot 2^{tw(G)+1}})$-time algorithm.

Also, the third result shows that the vector domination of planar graphs is subexponential FPT with respect to $k$, and it greatly improves the running time of existing $k^{O(k^{2})}n^{O(1)}$-time algorithm ([25]). It was shown in [5] that for the ordinary dominating set problem (equivalently, the vector domination (or multiple domination) with $d=(1,1,\ldots,1)$) in planar graphs, there is no $2^{o(\sqrt{k})}n^{O(1)}$-time algorithm unless the Exponential Time Hypothesis (i.e., the assumption that there is no $2^{o(n)}$-time algorithm for $n$-variable 3SAT [23]) fails. Hence, in this sense, our algorithm in third result (or the fourth results for the multiple domination) is optimal if $d^{*}$ is a constant.

The third and fourth results give subexponential fixed-parameter algorithms of the domination problems for planar graphs. It should be noted that the domination problems themselves do not have the bidimensionality, mentioned in the previous subsection, due to the existence of the vertices with demand $0$. Instead, by reducing irrelevant vertices, we obtain a similar inequality about the branchwidth and the solution size of the domination problems, which leads to the subexponential fixed-parameter algorithms.

The remainder of the paper is organized as follows. In Section 2, we introduce some basic notations and then explain the branch decomposition. Section 3 is the main part of the paper, and presents our dynamic programming based algorithms for the considered problems. Section 4 explains how we extend the algorithms of Section 3 to fixed-parameter algorithms for planar graphs.

A branch decomposition of a graph $G=(V,E)$ is defined as a pair $(T=(V_{T},E_{T}),\tau)$ such that (a) $T$ is a tree with $|E|$ leaves in which every non-leaf node has degree 3, and (b) $\tau$ is a bijection from $E$ to the set of leaves of $T$. Throughout the paper, we shall use node to denote an element in $V_{T}$ for distinguishing it from an element in $V$.

For an edge $f$ in $T$, let $T_{f}$ and $T-T_{f}$ be two trees obtained from $T$ by removing $f$, and $E_{f}$ and $E-E_{f}$ be two sets of edges in $E$ such that $e\in E_{f}$ if and only if $\tau(e)$ is included in $T_{f}$. The order function $w:E(T)\to 2^{V}$ is defined as follows: for an edge $f$ in $T$, a vertex $v\in V$ belongs to $w(f)$ if and only if there exist an edge in $E_{f}$ and an edge in $E-E_{f}$ which are both incident to $v$. The width of a branch decomposition $(T,\tau)$ is $\max\{|w(f)|\mid f\in E_{T}\}$, and the branchwidth of $G$, denoted by $bw(G)$, is the minimum width over all branch decompositions of $G$.

In general, computing the branchwidth of a given graph is NP-hard [28]. On the other hand, Bodlaender and Thilikos [4] gave a linear time algorithm which checks in linear time whether the branchwidth of a given graph is at most $k$ or not, and if so, outputs a branch decomposition of minimum width, for any fixed $k$. Also, as shown in the following lemma, it is known that for planar graphs, it can be done in polynomial time for any given $k$, where a graph is called planar if it can be drawn in the plane without generating a crossing by two edges.

Here, we introduce the following basic properties about branch decompositions, which will be utilized in the subsequent sections.

In this subsection, we consider the vector domination problem, and show the following theorem.

Below, for proving this theorem, we will give a dynamic programming algorithm for finding a minimum vector dominating set in $G$ in $O((d^{*}+2)^{b}\{(d^{*}+1)^{2}+1\}^{b/2}m)$ time, based on a branch decomposition of $G$.

Let $(T^{\prime},\tau)$ be a branch decomposition of $G=(V,E)$ with $b$, and $w^{\prime}:E(T^{\prime})\to 2^{V}$ be the corresponding order function. Let $T$ be the tree from $T^{\prime}$ by inserting two nodes $r_{1}$ and $r_{2}$, deleting one arbitrarily chosen edge $(x_{1},x_{2})\in E(T^{\prime})$, adding three new edges $(r_{1},r_{2})$, $(x_{1},r_{2})$, and $(x_{2},r_{2})$; namely, $T=(V(T^{\prime})\cup\{r_{1},r_{2}\},E(T^{\prime})\cup\{(r_{1},r_{2}),(x_{1},r_{2}),(x_{2},r_{2})\}-\{(x_{1},x_{2})\})$. Here, we regard $T$ with a rooted tree by choosing $r_{1}$ as a root. Let $w(f)=w^{\prime}(f)$ for every $f\in E(T)\cap E(T^{\prime})$, $w(x_{1},r_{2})=w(x_{2},r_{2})=w^{\prime}(x_{1},x_{2})$, and $w(r_{1},r_{2})=\emptyset$.

Let $f=(y_{1},y_{2})\in E$ be an edge in $T$ such that $y_{1}$ is the parent of $y_{2}$. Let $T(y_{2})$ be the subtree of $T$ rooted at $y_{2}$, $E_{f}=\{e\in E\mid\tau(e)\in V(T(y_{2}))\}$, and $G_{f}$ be the subgraph of $G$ induced by $E_{f}$. Note that $w(f)\subseteq V(G_{f})$ holds, since each vertex in $w(f)$ is an end-vertex of some edge in $E_{f}$ by definition of the order function $w$. In the following, each vertex $v\in w(f)$ will be assigned one of the following $d(v)+2$ colors

$$\{\top,0,1,2,\ldots,d(v)\}.$$

The meaning of the color of a vertex $v$ is as follows: for a vertex set (possibly, a vector dominating set) $D$,

• •

  $\top$ means that $v\in D$.

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  $i\in\{0,1,\ldots,d(v)\}$ means that $v\notin D$ and $|N_{G_{f}}(v)\cap D|\geq d(v)-i$.

Notice that a vertex colored by $i>0$ may need to be dominated by some vertices in $V-V(G_{f})$ for the feasibility. Given a coloring $c\in\{\top,0,1,2,\ldots,d^{*}\}^{|w(f)|}$, let $D_{f}(c)\subseteq V(G_{f})$ be a vertex set with the minimum cardinality satisfying the following (1)–(3), where $c(v)$ denotes the color assigned to a vertex $v\in V$:

	$c(v)=\top$ if and only if $v\in D_{f}(c)\cap w(f)$.		(1)
	If $c(v)=i$, then $v\in w(f)-D_{f}(c)$ and $|N_{G_{f}}(v)\cap D_{f}(c)|\geq d(v)-i$.		(2)
	$|N_{G_{f}}(v)\cap D_{f}(c)|\geq d(v)$ holds for every vertex $v\in V(G_{f})-w(f)-D_{f}(c)$.		(3)

Our dynamic programming algorithm proceeds bottom-up in $T$, while computing $A_{f}(c)$ for all $c\in\{\top,0,1,2,\ldots,d^{*}\}^{|w(f)|}$ for each edge $f$ in $T$. We remark that $A_{(r_{1},r_{2})}(c)$ is the cardinality of a minimum vector dominating set, because $w(r_{1},r_{2})=\emptyset$ and $G_{(r_{1},r_{2})}=G$. The algorithm consists of two types of procedures: one is for leaf edges and the other is for non-leaf edges, where a leaf edge denotes an edge incident to a leaf of $T$.

For a fixed $c$, we need to check if (1) – (3) hold. This can be done in $O(|w(f)|)$ time. Hence, this step takes $O((d^{*}+2)^{|w(f)|}|w(f)|)$ time.

We say that a coloring $c\in\{\top,0,1,2,\ldots,d^{*}\}^{|w(f)|}$ of $w(f)$ is formed from a coloring $c_{1}$ of $w(f_{1})$ and a coloring $c_{2}$ of $w(f_{2})$ if the following (P1)–(P5) hold.

(a) If $v\in X_{1}\cup X_{3}$, then $c_{1}(v)=\top$.
(b) If $v\in X_{2}\cup X_{3}$, then $c_{2}(v)=\top$.

As we will show in Lemmas 3 and 4, there exist a coloring $c_{1}$ of $w(f_{1})$ and a coloring $c_{2}$ of $w(f_{2})$ forming $c$ such that $D_{f_{1}}(c_{1})\cup D_{f_{2}}(c_{2})$ satisfies (1)–(3) and $|D_{f_{1}}(c_{1})\cup D_{f_{2}}(c_{2})|=A_{f}(c)$. Namely, we have

$$A_{f}(c)=\min\{|A_{f_{1}}(c_{1})|+|A_{f_{2}}(c_{2})|-|D_{c_{1},c_{2}}\cap(X_{3}\cup X_{4})|\mid\mbox{ $c_{1},c_{2}$ forms $c$}\}.$$

Thus, for all colorings $c\in\{\top,0,1,2,\ldots,d^{*}\}^{|w(f)|}$, we can compute $A_{f}(c)$ from the information of $f_{1}$ and $f_{2}$. By repeating these procedure bottom-up in $T$, we can find a minimum vector dominating set in $G$.

Here, for a fixed $c$, we analyze the time complexity for computing $A_{f}(c)$. Let $D_{c}=\{v\in w(f)\mid c(v)=\top\}$, $x_{j}=|X_{j}|$ for $j=1,2,3,4$, $z_{3}=|X_{3}-D_{c}|$, and $z_{4}$ be the number of vertices in $X_{4}$ not colored by $\top$. The number of pairs of a coloring $c_{1}$ of $w(f_{1})$ and a coloring $c_{2}$ of $w(f_{2})$ forming $c$ is at most

$$(d^{*}+1)^{z_{3}}\sum_{z_{4}=0}^{x_{4}}{x_{4}\choose z_{4}}(d^{*}+1)^{z_{4}}(d^{*}+1)^{z_{4}}$$

since the number of pairs $(i_{1},i_{2})$ in (P4) or (P5) is at most $d^{*}+1$ for each vertex in $X_{3}-D_{c}$ or each vertex in $X_{4}$ not colored by $\top$.

Hence, for an edge $f$, the number of pairs forming $c$ is at most

$$\begin{array}[]{l}(d^{*}+2)^{x_{1}+x_{2}}\sum_{z_{3}=0}^{x_{3}}{x_{3}\choose z_{3}}(d^{*}+1)^{z_{3}}(d^{*}+1)^{z_{3}}\sum_{z_{4}=0}^{x_{4}}{x_{4}\choose z_{4}}(d^{*}+1)^{z_{4}}(d^{*}+1)^{z_{4}}\\[5.69046pt] =(d^{*}+2)^{x_{1}+x_{2}}\{(d^{*}+1)^{2}+1\}^{x_{3}+x_{4}}\end{array}$$

in total. Now we have $x_{1}+x_{2}+x_{3}\leq b$, $x_{1}+x_{3}+x_{4}\leq b$, and $x_{2}+x_{3}+x_{4}\leq b$ (recall that $b$ is the width of $(T^{\prime},\tau)$). By considering a linear programming problem which maximizes $(x_{1}+x_{2})\log(d^{*}+2)+(x_{3}+x_{4})\log\{(d^{*}+1)^{2}+1\}$ subject to these inequalities, we can observe that $(d^{*}+2)^{x_{1}+x_{2}}\{(d^{*}+1)^{2}+1\}^{x_{3}+x_{4}}$ attains the maximum when $x_{1}=x_{2}=x_{4}=b/2$ and $x_{3}=0$. Thus, it takes in $O((d^{*}+2)^{b}\{(d^{*}+1)^{2}+1\}^{b/2})$ time to compute $A_{f}(c)$ for all colorings $c$ of $w(f)$.

Since $|E(T)|=O(m)$ and the initialization step takes $O((d^{*}+2)^{b}m)$ time in total, we can obtain $A_{(r_{1},r_{2})}(c)$ in $O(((d^{*}+2)^{b}\{(d^{*}+1)^{2}+1\}^{b/2}m)$time.