Distributed Domination on Graph Classes of Bounded Expansion

TL;DR

This paper introduces a distributed approximation algorithm for the distance-$r$ dominating set problem on bounded expansion graph classes, achieving efficient computation in the ext{CONGEST}_{BC} model and improving connected dominating set approximations.

Contribution

It presents a novel distributed algorithm for approximating distance-$r$ dominating sets on bounded expansion graphs, utilizing sparse neighborhood covers and extending to connected dominating sets.

Findings

Algorithm computes $r$-neighborhood covers in $ ext{O}(r^2 ext{log} n)$ rounds.

Provides a constant factor approximation for connected dominating sets in bounded expansion graphs.

Achieves efficient distributed computation in the ext{CONGEST}_{BC} and ext{LOCAL} models.

Abstract

We provide a new constant factor approximation algorithm for the (connected) distance-$r$ dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance-$r$ dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $\mathcal{O}(r^2 \log n)$ communication rounds. Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an $r$-neighborhood cover of radius~$2r$ and overlap $f(r)$ on every class of bounded expansion in $\mathcal{O}(r^2 \log n)$ communication rounds for some function~$f$.% in the $\mathcal{CONGEST}_{\mathrm{BC}}$ model. Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set into a constantly larger connected distance-$r$ dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al.\ gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the $\mathcal{LOCAL}$ model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.'s algorithm.

Equations

$${\mathrm{wcol}}_r(G)=\underset{L\in \Pi (G)}{\min }\underset{v\in V(G)}{\max }|{\mathrm{WReach}}_r[G,L,v]|.$$

$$X_v=\{w\in V(G)\text{leavevmode}:\text{leavevmode}v\in {\mathrm{WReach}}_{2r}[G,L,w]\}.$$

$$R_v:=\{w\in X_v\text{leavevmode}:\text{leavevmode}v=\min {\mathrm{WReach}}_r[G,L,w]\}.$$

$$D:=\{\min {\mathrm{WReach}}_r[G,L,w]\text{leavevmode}:\text{leavevmode}w\in V(G)\}=\{v\in V(G)\text{leavevmode}:\text{leavevmode}{R}_v\ne \varnothing \}$$

$$|D|=|\bigcup _{\stackrel{v\in V(G)}{R_v\ne \varnothing }}\{v\}|\le \sum _{\stackrel{v\in V(G)}{R_v\ne \varnothing }}|M\cap X_v|\le c(r)\cdot |M|.$$

$$D:=\{v\in V(G)\text{leavevmode}:\text{leavevmode}v=\min {\mathrm{WReach}}_r[G,L,w]\text{ for some }w\in V(G)\},$$

$$B(v):=\{w\in V(G)\text{leavevmode}:\text{leavevmode}P(v,w){\le }_{\mathrm{lex}}P(u,w)\text{ for all }u\in D,u\ne v\}.$$

Full text

Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an $r$-neighborhood cover of radius $2r$ and overlap $f(r)$ on every class of bounded expansion in $\mathcal{O}(r^{2}\log n)$ communication rounds for some function $f$.

Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set into a constantly larger connected distance-$r$ dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al. gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the $\mathcal{LOCAL}$ model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.’s algorithm.

1 Introduction and contributions

The Dominating Set and Connected Dominating Set problems are two of the most well-studied problems in algorithms and combinatorics [28]. Recall that a subset $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex of $G$ is either in $D$ or adjacent to a vertex in $D$, and that a dominating set is a connected dominating set if it induces a connected subgraph.

The Dominating Set problem, which aims at finding a minimum size dominating set in a graph is NP-complete in general [31], and even so on planar graphs of maximum degree $3$ (cf. [GT2] in [23]). The simple greedy algorithm—which at each step adds a vertex dominating the largest number of non-dominated vertices—achieves an approximation ratio 222Note that these results are for the Set Cover problem, which however reduces to the Dominating Set problem by an approximation preserving reduction and, in fact, the two problems achieve exactly the same approximation ratio [30]. of $\ln n-\ln\ln n+\Theta(1)$ on graphs of order $n$ [15, 39], and no better approximation ratio can be achieved in general under standard complexity theoretic assumptions [4, 6, 12, 20, 22, 40, 50].

Recall that a class $\mathcal{C}$ (closed under taking subgraphs) has bounded expansion if, for every integer $r$, the average degree of all graphs having their $r$-subdivision (that is the graph obtained by replacing each edge by a path of length $r+1$) in $\mathcal{C}$ is bounded by some constant $C(r)$ [42, 43, 44, 45]. Not only do many familiar classes of sparse graphs, such as planar graphs and graphs with excluded (topological) minors have bounded expansion, but so do many geometrically defined graphs [47, 27], and experimental evidence that real world complex networks have bounded expansion is given in [19]. Also, widely used random models of sparse graphs, like the Configuration Model [41] and the Chung–Lu Model [14] with specified asymptotic degree sequences generate graphs that asymptotically almost surely belong to a bounded expansion class determined by the parameters of the model [19].

In [21] it is proved that in linear time one can compute a constant factor approximation on classes with bounded expansion for a generalization of the Dominating Set, the Distance-$r$ Dominating Set problem, which consists in finding in an input graph $G$ a minimum size subset $D$ of vertices, such that every vertex of $G$ is at distance at most $r$ from a vertex in $D$. This problem (also known as the $(k,r)$-center problem) has been extensively studied in the literature.

Contribution 1. We present a new approximation algorithm for the Distance-$r$ Dominating Set problem on classes of bounded expansion which improves the approximation ratio achieved by the algorithm of [21]. Our algorithm can be implemented in linear time on any class of bounded expansion. A key feature of our algorithm is that it is tailored to be executed in a distributed setting.

There has been lots of effort to approximate the Dominating Set problem with distributed algorithms, however, similar hardness results also apply to distributed algorithms. It was shown in [34] that in $t$ communication rounds the Dominating Set problem on $n$-vertex graphs of maximum degree $\Delta$ can only be approximated within factor $\Omega(n^{c/t^{2}})$ and $\Omega(\Delta^{c^{\prime}/t})$, where $c$ and $c^{\prime}$ are constants. This implies that, in general, to achieve a constant approximation ratio, every distributed algorithm requires at least $\Omega(\sqrt{\log n})$ and $\Omega(\log\Delta)$ communication rounds. Kuhn et al. [34] also provides an approximation algorithm on general graphs, which achieves a $(1+\varepsilon)\ln\Delta$-approximation in $\mathcal{O}(\log(n)/\varepsilon)$ rounds for any $\varepsilon>0$. Ghaffari et al. [24] provide a polylog-time distributed approximation scheme for covering and packing integer linear programs. In particular, based on their techniques one can compute a $(1+\varepsilon)$-approximation for dominating sets and distance-$r$ dominating sets in $\mathcal{O}(\mathit{poly}(\log n/\varepsilon))$ rounds in the $\mathcal{LOCAL}$ model. Observe however that this result requires learning polylog-neighborhoods and solving the dominating set problem optimally on these neighborhoods. In particular, the algorithm cannot be carried out in the $\mathcal{CONGEST}$ model (in this number of steps).

For graphs of arboricity $a$ there exists a forest decomposition algorithm achieving a factor $\mathcal{O}(a^{2})$-approximation in randomized time $\mathcal{O}(\log n)$, and a deterministic $\mathcal{O}(a\log\Delta)$ approximation algorithm requiring $\mathcal{O}(\log\Delta)$ rounds [38]. Given any $\delta>0$, $(1+\delta)$-approximations of a maximum independent set, of a maximum matching, and of a minimum dominating set can be computed in $\mathcal{O}(\log^{*}n)$ rounds in planar graphs [16], which is asymptotically optimal [37]. A constant factor approximation on planar graphs [36, 57] and on graphs of bounded genus [5] can be computed locally in a constant number of communication rounds. In terms of lower bounds, it was shown that there is no deterministic local algorithm (constant-time distributed graph algorithm) that finds a $(7-\varepsilon)$-approximation of a minimum dominating set on planar graphs, for any positive constant $\varepsilon$ [29].

Observe that the above algorithms for restricted graph classes cannot be directly employed to obtain good approximations for the Distance-$r$ Dominating Set problem, as all structural information which is used in the algorithms may be lost when building the $r$-transitive closure of the graph. The distributed algorithms of [35, 49] find distance-$r$ dominating sets of size $\mathcal{O}(n/r)$ in time $\mathcal{O}(r\cdot\log^{*}n)$, without any relation to the size of an optimal distance-$r$ dominating set. In very restrictive settings, e.g., in trees [54] or in star-split graphs [56] better solutions are known.

The Distance-$r$ Dominating Set problem is closely related to the problem of covering local neighborhoods in a graph by connected clusters of small radius. An $r$-neighborhood cover [9] is a set $\mathcal{X}$ of vertex sets $X\subseteq V(G)$ such that for each vertex $v\in V(G)$ there is a set $X\in\mathcal{X}$ with $N_{r}[v]\subseteq X$. We are interested in covers of small radius, that is, $\mathrm{rad}(G[X])$ shall be small for all $X\in\mathcal{X}$ and small degree, that is, every vertex $v\in V(G)$ shall lie in only a few clusters.

Sparse covers have many applications such as distance coordinates, routing with succinct routing tables [2, 9], mobile user tracking [9], resource allocation [7], synchronisation in distributed algorithms [8], and many more. Every graph admits an $r$-neighborhood cover of radius at most $2r-1$ and degree at most $2r\cdot n^{1/r}$ [9] and asymptotically, this cannot be improved [53]. Better covers are known to exist, e.g., for planar graphs [13] and for classes that exclude a minor [3]. In particular, the construction of [1] provides $r$-neighborhood covers of radius $\mathcal{O}(t^{2}r)$ and degree $2^{\mathcal{O}(t)}\cdot t!$ for graphs that exclude $K_{t}$ as a minor. It follows from a construction in [26] that classes of bounded expansion admit $r$-neighborhood covers of radius at most $2r$ and degree at most $f(r)$ for some function $f$.

Contribution 2. We show that the algorithm of [26] for constructing sparse $r$-neighborhood covers on classes of bounded expansion can be implemented in the $\mathcal{CONGEST}_{\textsc{BC}}$ model of distributed computing in $\mathcal{O}(r^{2}\log n)$ communication rounds. Based on this construction, we show that our newly proposed algorithm for the Distance-$r$ Dominating Set problem can be implemented in the $\mathcal{CONGEST}_{\textsc{BC}}$ model in $\mathcal{O}(r^{2}\log n)$ communication rounds on any class of graphs of bounded expansion. Our result is based on a routing scheme presented by Nešetřil and Ossona de Mendez in [46], which in turn is based on an iterative application of an algorithm of Barenboim and Elkin [11].

While in a sequential setting one can trivially connect the vertices of a (distance-$r$) dominating set along a spanning tree to obtain a connected (distance-$r$) dominating set of small size, creating such connections is a non-trivial task in the distributed setting. Several algorithms were proposed to compute connected dominating sets in general graphs [17, 18, 34, 51, 52, 58]. We also refer to these papers for applications of connected dominating sets for distributed computing and routing. All lower bounds for the Dominating Set problem hold all the more so for the Connected Dominating Set problem. In particular, none of the above algorithms computes a constant factor approximation of a minimum connected dominating set in a sub-linear number of communication rounds. To our knowledge, there is no distributed algorithm to compute a constant factor approximation to the Connected (Distance-$r$) Dominating Set problem on restricted graph classes.

Contribution 3. We show how to extend our algorithm for the Distance-$r$ Dominating Set problem to compute a constant factor approximation for the Connected Distance-$r$ Dominating Set problem. We hence prove that there exists a constant factor approximation algorithm for the Connected Distance-$r$ Dominating Set problem which works in the $\mathcal{CONGEST}_{\textsc{BC}}$ model in $\mathcal{O}(r^{2}\log n)$ communication rounds on any class of graphs of bounded expansion.

Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set $D$ into a connected distance-$r$ dominating set of size at most $c(r)\cdot|D|$, for some small constant $c(r)$ depending only on $r$ and the class under consideration. This new algorithm can be implemented in $3r+1$ communication rounds in the $\mathcal{LOCAL}$ model. In combination with the algorithm of Lenzen et al. [36] we obtain a constant factor approximation algorithm for the Connected Dominating Set problem on planar graphs in a constant number of communication rounds in the $\mathcal{LOCAL}$ model (the constant $c(1)$ which we need here is $6$). A similar result follows for graphs of bounded genus by combining our new algorithm with an algorithm of [5].

2 Preliminaries

Graphs.

In this paper, we consider finite, undirected simple graphs. For a graph $G$, we write $V(G)$ for the vertex set of $G$ and $E(G)$ for its edge set. A path of length $\ell$ in $G$ is a subgraph $P\subseteq G$ with vertex set $V(P)=\{v_{1},\ldots,v_{\ell+1}\}$ and edge set $E(P)=\{\{v_{i},v_{i+1}\}\leavevmode\nobreak\ :\leavevmode\nobreak\ 1\leq i<\ell\}$. The path $P$ connects its endpoints $v_{1}$ and $v_{\ell+1}$. The distance between two vertices $u,v\in V(G)$, denoted $\mathrm{dist}(u,v)$, is the minimum length of a path that connects $u$ and $v$ or $\infty$ if no such path exists. For $v\in V(G)$, we write $N_{r}[v]$ for the closed $r$-neighborhood of $v$, that is $N_{r}[v]=\{u\in V(G)\leavevmode\nobreak\ :\leavevmode\nobreak\ \mathrm{dist}(u,v)\leq r\}$. Note that we allow paths of length [math], so $N_{r}[v]$ always contains $v$ itself. For a set $A\subseteq V(G)$, we write $N_{r}[A]$ for $\bigcup_{v\in A}N_{r}[v]$. The radius of a connected graph $G$ is the minimum number $\mathrm{rad}(G)$ such that there is a vertex $v\in V(G)$ with $N_{\mathrm{rad}(G)}[v]=V(G)$.

The arboricity of a graph is the minimum number of spanning forests that partition its edge set. The arboricity of a graph is within factor $2$ of its degeneracy. For a set $X\subseteq V(G)$ we write $G[X]$ for the subgraph of $G$ induced by $X$. For $k\in\mathbb{N}$, $G$ is $k$-degenerate if for each $X\subseteq V(G)$ the graph $G[X]$ contains a vertex of degree at most $k$. If an $n$-vertex graph $G$ is $k$-degenerate, then $G$ contains at most $k\cdot n$ edges.

We assume that all graphs are represented by adjacency lists so that the total size of a graph representation is linear in the number of edges and vertices.

If $G$ is $k$-degenerate, then in linear time we can order the vertices as $v_{1},\ldots,v_{n}$ such that every vertex $v_{i}$ has at most $k$ smaller neighbours $v_{j_{1}},\ldots,v_{j_{k}}$, $j_{\ell}<i$ for $\ell\in\{1,\ldots,k\}$. In the same time complexity we can order all adjacency lists consistently with the order. For the sequential model in Section 3 we assume that the identifiers of the vertices occupy constant space while in the distributed model we assume $\log n$-bit identifiers.

Distance - r dominating sets.

For an integer $r$, a distance - $r$ dominating set in a graph $G$ is a set $M\subseteq V(G)$ such that $N_{r}[M]=V(G)$. A distance-$1$ dominating set is simply called a dominating set.

Distributed system model.

The clients of a network are modelled as the vertices $V(G)$ of a graph $G$, its communication links are represented by the edges $E(G)$ of the graph. Each client has a unique identifier (id) of size $\log n$ where $n\coloneqq|V(G)|$ is the order of the graph known to every vertex. Communication is synchronous and reliable. In each round, each vertex $v\in V(G)$ may send a (different) message to each of its neighbors $w\in N_{1}[v]$ (the vertex specifies which message is sent to which neighbor) and receives all messages from its neighbors. In the $\mathcal{LOCAL}$ model, messages may have arbitrary size, in the $\mathcal{CONGEST}$ model, messages may have size $\mathcal{O}(\log n)$. In the $\mathcal{CONGEST}_{\textsc{BC}}$ model, every vertex may only broadcast the same message of size $\mathcal{O}(\log n)$ to all its neighbors. After sending and receiving messages, every client may perform arbitrary finite computations. The complexity of a distributed algorithm is its number of communication rounds. The network graph also represents the graph problem that we are trying to solve, e.g., the Distance-$r$ Dominating Set instance. At termination, each vertex must output whether it is part of the distance-$r$ dominating set or not, and these outputs must define a valid solution of the problem. We refer to [48] for more background.

Bounded expansion classes.

A graph $H$ with vertex set $V(H)=\{v_{1},\ldots,v_{n}\}$ is a minor of $G$, written $H\preccurlyeq G$, if there are pairwise disjoint connected subgraphs $H_{1},\ldots,H_{n}\subseteq G$, called branch sets, such that if $\{v_{i},v_{j}\}\in E(H)$, then there are vertices $u_{i}\in V(H_{i})$ and $u_{j}\in V(H_{j})$ with $\{u_{i},u_{j}\}\in E(G)$. We call $(H_{1},\ldots,H_{n})$ a minor model of $H$ in $G$. For $r\in\mathbb{N}$, the graph $H$ is a depth-$r$ minor of $G$, denoted $H\preccurlyeq_{r}G$, if there is a minor model $(H_{1},\ldots,H_{n})$ of $H$ in $G$ such that each $H_{i}$ has radius at most $r$. We write $d(H)$ for the average degree of $H$, that is, for the number $2\lvert E(H)\rvert/\lvert V(H)\rvert$. A class $\mathcal{C}$ of graphs has bounded expansion if there is a function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ such that for all $r\in\mathbb{N}$ and all graphs $H$ and $G\in\mathcal{C}$, if $H\preccurlyeq_{r}G$, then $d(H)\leq f(r)$. Observe that every $n$-vertex graph from a bounded expansion class is $f(0)$-degenerate and hence has at most $f(0)\cdot n$ many edges (depth-[math] minors of $G$ are its subgraphs).

Generalized colouring numbers.

Let $G$ be a graph. A linear order $L$ of $V(G)$ is a reflexive, anti-symmetric, transitive total binary relation $L\subseteq V(G)\times V(G)$. In the following, we will write $u\leq_{L}v$ instead of $(u,v)\in L$. We write $\Pi(G)$ for the set of all linear orders on $V(G)$. Let $r\in\mathbb{N}$ and let $u,v\in V(G)$. Vertex $u$ is weakly $r$-reachable from vertex $v$ with respect to a linear order $L\in\Pi(G)$ if there exists a path $P$ of length at most $r$ between $u$ and $v$ such that $u$ is minimum among the vertices of $P$ (with respect to $L$). Let $\mathrm{WReach}_{r}[G,L,v]$ be the set of vertices that are weakly $r$-reachable from $v$ with respect to $L$. Note that $v\in\mathrm{WReach}_{r}[G,L,v]$. The weak $r$-colouring number $\mathrm{wcol}_{r}(G)$ of $G$ is defined as

[TABLE]

The generalized colouring numbers were introduced by Kierstead and Yang in the context of colouring games and marking games on graphs [32], and received much attention as a measure for uniform sparseness in graphs, in particular, they can be used to characterize classes of bounded expansion.

Theorem 1** (Zhu [59]).**

*A class $\mathcal{C}$ of graphs has bounded expansion if and only if there is a function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ such that $\mathrm{wcol}_{r}(G)\leq f(r)$ for all $r\in\mathbb{N}$. *

Bounds for several restricted classes such as graphs of bounded tree-width, planar graphs or graphs with excluded (topological) minors, were provided in [25, 33, 55].

The weak colouring numbers can be well approximated in linear time as shown in [21].

Theorem 2** (Dvořák [21]).**

*Let $\mathcal{C}$ be a class of bounded expansion. There is a linear time algorithm and a function $d:\mathbb{N}\rightarrow\mathbb{N}$ which on input $G\in\mathcal{C}$ and $r\in\mathbb{N}$ computes in linear time a linear order of $V(G)$ witnessing that $\mathrm{wcol}_{r}(G)\leq d(r)$. *

The next theorem is implicit in [46, Section 4.4] and shows that we can compute these orders also in the distributed setting.

Theorem 3** (Nešetřil and Ossona de Mendez [46]).**

*Let $\mathcal{C}$ be a class of bounded expansion and let $r\in\mathbb{N}$. There is a constant $d(r)$ such that one can compute for every $G\in\mathcal{C}$ in $\mathcal{O}(r^{2}\log n)$ communication rounds in the $\mathcal{CONGEST}_{\textsc{BC}}$ model an order of $V(G)$ witnessing that $\mathrm{wcol}_{r}(G)\leq d(r)$. *

The procedure described in theorem 3 uses an algorithm by Barenboim and Elkin [11] which computes an orientation of degenerate graphs. For this, we must assume that all vertices know the order $n$ of the input graph. The order is represented by assigning every vertex a class-id, which together with the unique vertex-id induces a total order of $V(G)$. We remark that (though not explicitly stated) this order can be obtained as a by-product of the procedure $\mathrm{orient}(z,C)$ described in [46, Section 4.4].

In fact, it suffices if the vertices know a polynomial approximate of $n$ as this changes $\log n$ only by a constant factor. In the following, we will assume that the exact value of $n$ is available to avoid unnecessary complication.

Sparse neighborhood covers.

Let $G$ be a graph. For $r\in\mathbb{N}$, an $r$-neighborhood cover of $G$ is a set $\mathcal{X}$ of subsets $X\subseteq V(G)$, called the clusters of $\mathcal{X}$, such that for each $v\in V(G)$ there is some $X\in\mathcal{X}$ with $N_{r}(v)\subseteq X$. The radius of $\mathcal{X}$ is the maximum radius of the graph induced by a cluster $X\in\mathcal{X}$. Note that in every $r$-neighborhood cover of bounded radius each of the clusters induces a connected subgraph of $G$. The degree $d_{\mathcal{X}}(v)$ of a vertex $v\in V(G)$ with respect to $\mathcal{X}$ is the number of clusters that contain $v$. The degree of $\mathcal{X}$ is the maximum degree $d_{\mathcal{X}}(v)$ over all vertices $v\in V(G)$. The generalized colouring numbers can be used to construct sparse neighborhood covers.

We fix a number $r\in\mathbb{N}$ for the remainder of the paper. For a vertex $v\in V(G)$, let $X_{v}$ be the set of the vertices $w$ such that $v\in\mathrm{WReach}_{2r}[G,L,w]$.

Theorem 4** (Grohe et al. [26]).**

*Let $G$ be a graph and let $c,r\in\mathbb{N}$. Let $L$ be an order witnessing that $\mathrm{wcol}_{2r}(G)\leq c$. Then the collection $\mathcal{X}=\{X_{v}\leavevmode\nobreak\ :\leavevmode\nobreak\ v\in V(G)\}$ is an $r$-neighborhood cover of $G$ of radius $2r$ and degree $c$. *

Hence, by combining theorem 1 and theorem 4, we obtain $r$-neighborhood covers of radius at most $r$ and degree at most $f(r)$ for every class of bounded expansion. We will show in section 4 how to compute the $r$-neighborhood covers presented in theorem 4 in a distributed setting.

3 Approximating dominating sets

Our first result shows how to (sequentially) compute good distance-$r$ dominating sets in any fixed class of bounded expansion. The remainder of this section is devoted to the proof of the following theorem.

Theorem 5**.**

*For every class $\mathcal{C}$ of bounded expansion there is a function $c\colon\mathbb{N}\rightarrow\mathbb{N}$ and a linear time algorithm which on input $G\in\mathcal{C}$ and $r\in\mathbb{N}$ computes an order $L\in\Pi(G)$ witnessing $\mathrm{wcol}_{2r}(G)\leq c(r)$ and a $c(r)$-approximation of a minimum distance-$r$ dominating set of $G$. *

Our result improves the following result of Dvořák [21] who proved that there exists a $c(r)^{2}$-approximation, where $c(r)$ is the constant in theorem 5.

Recall the definition of $X_{v}$ (see section 2):

[TABLE]

We define for a fixed vertex $v\in V(G)$,

[TABLE]

Lemma 6**.**

*For all vertices $v\in V(G)$ and for all $w\in R_{v}$ we have $N_{r}[w]\subseteq X_{v}$. *

Proof.

Let $w\in R_{v}$. Observe first that $v$ is the minimum element in $N_{r}[w]$. Now let $u\in N_{r}[w]$. Then there is a path of length at most $r$ from $u$ to $w$ with all vertices in $N_{r}[w]$ and a path of length at most $r$ from $w$ to $v$ again with all vertices in $N_{r}[w]$. Hence there is a path of length at most $2r$ from $u$ to $v$ with all vertices in $N_{r}[w]$. As $v$ is minimal in $N_{r}[w]$, this path witnesses that $v\in\mathrm{WReach}_{2r}[G,L,u]$, and hence by definition of $X_{v}$ it holds that $u\in X_{v}$. $\square$

Proof. (Proof of theorem 5)

We claim that the set

[TABLE]

is a $c(r)$-approximation of a minimum distance-$r$ dominating set. Obviously, $D$ is a distance-$r$ dominating set of $V(G)$, as every vertex $w$ is dominated by $\min\mathrm{WReach}_{r}[G,L,w]$. It remains to show that we achieve the claimed approximation ratio.

For $v\in V(G)$, let $X_{v}$ and $R_{v}$ be as above. Let $\mathcal{X}$ be the collection $\{X_{v}\leavevmode\nobreak\ :\leavevmode\nobreak\ v\in V(G)\}$ as in Theorem 4. Then $\mathcal{X}$ is an $r$-neighborhood cover of degree $c(r)$ and by Lemma 6, for $w\in R_{v}$, we have $N_{r}[w]\subseteq X_{v}$.

Let $M$ be a minimum distance-$r$ dominating set in $G$. As every $w\in V(G)$ can be distance-$r$ dominated only from $N_{r}[w]\subseteq X_{v}$, it follows that if $w\in R_{v}$, then $M\cap X_{v}\neq\emptyset$.

Hence, as every vertex appears in at most $c(r)$ clusters, it holds that

[TABLE]

We finally show how to compute $D$ in linear time. We assume that $G$ is stored in memory by $n$ adjacency lists. We first use the linear time approximation algorithm for the weak coloring numbers from theorem 2 to compute an order $L$, which will be represented in a way such that one can iterate through the vertices along $L$ in $\mathcal{O}(|V(G)|)$ time and such that a comparison $u<_{L}w$ for every pair $u,w\in V(G)$ takes $\mathcal{O}(1)$ time. Here we assume that vertices are equipped with constant size identifiers representing the order. The algorithm is given as Algorithm 1.

In the first step (3 of Algorithm 1 and Algorithm 2), we ensure in linear time that every adjacency list is sorted increasingly with respect to $L$. Algorithm 2 iterates through the vertices of $G$ in order $L$ starting from the least vertex (such that the next vertex can always be found in constant time) and thus has running time $\mathcal{O}(m)$. As graphs of bounded expansion are degenerate, we have $m\in\mathcal{O}(n)$.

Now Algorithm 1 iterates through the vertices of $G$ starting with the least element $v_{1}$ along $L$. For every $v_{i}\in V(G)$, it uses Algorithm 3 to compute the set of vertices that are bigger than $v_{i}$ and are dominated by $v_{i}$. If such a vertex is not dominated by a vertex smaller than $v_{i}$, it serves as a vertex $w$ in the definition of $D$ (see eq. 2). Indeed, $v_{i}\in\mathrm{WReach}_{r}[G,L,w]$ because $w$ was found by a breadth-first search from $v$ restricted to vertices greater than $v$ with respect to $L$ and to distances at most $r$. On the other hand, if not $v$ was the minimum vertex in $\mathrm{WReach}_{r}[G,L,w]$ but, say, $u$, then $w$ would be dominated by $u$ and added to the set $Dominated$ in the earlier iteration $j$ for $u=v_{j}$. Thus $v_{i}=\min\mathrm{WReach}_{r}[G,L,w]$ and $v_{i}$ is added to $D$.

Let us estimate the running time of Algorithm 1. Recall that 3 has linear running time. Note that every set $N_{i}$ computed in LABEL:{line:BFS} for a vertex $v_{i}$ is a subset of $X_{v_{i}}$ because Algorithm 3 restricts its search to vertices bigger than $v_{i}$ and to distances at most $r$. That is, if $w\in N_{v_{i}}$, then $v_{i}\in\mathrm{WReach}_{r}[G,L,w]\subseteq\mathrm{WReach}_{2r}[G,L,w]$ and thus $w\in X_{v_{i}}$. As every graph $G\in\mathcal{C}$ is $c(r)$-degenerate, every induced subgraph $H\subseteq G$ has at most $c(r)\cdot|V(H)|$ many edges, also the graph induced by $N_{i}$. When constructing $N_{i}$ in Algorithm 3 we will only visit vertices of $N_{i}$ and, for every $w\in N_{i}$ at most one vertex in its adjacency list that is not in $N_{i}$. This can be achieved if 8 of Algorithm 3 is implemented as an iteration through $A(w)$ starting from the biggest vertex and stopping if a vertex $u\in A(w)$ with $u<_{L}w$ is reached (recall that $A(w)$ sorted). Hence this search requires time at most $\mathcal{O}((c(r)+1)\cdot|N_{v_{i}}|)=\mathcal{O}(c(r)\cdot|X_{v_{i}}|)$. As every vertex $w$ appears in at most $c(r)$ clusters $X_{v}$, we obtain a running time of $\sum_{v\in V(G)}\mathcal{O}(c(r)\cdot|X_{v}|)=\mathcal{O}(c(r)^{2}\cdot n)$.

$\square$

Note that by theorem 1 the constant $c(r)$ in the theorem exists for every class of bounded expansion. Besides the improved approximation ratio, our algorithm is simpler than that of [21]. In particular, given an order $L\in\Pi(G)$, it can straightforwardly be implemented in a distributed way.

4 Distributed r-neighborhood covers and r-dominating sets

In this section we will show how to compute sparse $r$-neighborhood covers as described in theorem 4 and the Distance-$r$ Dominating Set of theorem 5 in a distributed setting.

In order to compute $r$-neighborhood covers according to theorem 4, we want to compute an order $L$ of $V(G)$ which witnesses that $\mathrm{wcol}_{2r}(G)\leq c\coloneqq c(2r)$. In the distributed setting, that means that every vertex $w$ learns its weak reachability set $\mathrm{WReach}_{2r}[G,L,w]$ and, for each $v\in\mathrm{WReach}_{2r}[G,L,w]$, a path within $X_{v}$ of length at most $2r$ from $w$ to $v$.

In order to find the distance-$r$ dominating set described in theorem 5, every vertex $w$ will choose as its dominator the vertex $\min\mathrm{WReach}_{r}[G,L,w]$ and send a message to that vertex along the stored path. (Note that we computed the order $L$ for the parameter $2r$, but are using it for $r$.) Even if all vertices send their messages at once, no vertex will have to forward more than $c$ messages.

First, using theorem 3, we compute for a given input graph $G$ an order $L$ witnessing that $\mathrm{wcol}_{2r}(G)\leq c$ using $\mathcal{O}(r^{2}\log n)$ communication rounds. Note that the number of rounds does not depend on $c$, which influences only the size of messages. The latter is $\mathcal{O}(c^{2}\cdot r\cdot\log n)$, which implicitly follows from theorem 3 in [46]. We show that every vertex can learn its weak reachability set as well as a routing scheme which preserves short distances. Recall from Equation 1 on Equation 1 that $X_{v}$ is defined as $X_{v}=\{w\in V(G):v\in\mathrm{WReach}_{2r}[G,L,w]\}$.

Lemma 7**.**

*Let $\mathcal{C}$ be a class of bounded expansion and $r\in\mathbb{N}$. There is a constant $c=c(2r)$ such that for every $G\in\mathcal{C}$ there is a linear order $L$ on $V(G)$ such that $|\mathrm{WReach}_{2r}[G,L,w]|\leq c$ for all $w\in V(G)$ and in $\mathcal{O}(r^{2}\cdot\log n)$ communication rounds (in $\mathcal{CONGEST}_{\textsc{BC}}$) every vertex $w$ can learn $\mathrm{WReach}_{2r}[G,L,w]$ and for each $v\in\mathrm{WReach}_{2r}[G,L,w]$ a path $P_{v,w}$ of length at most $2r$ from $w$ to $v$, which is a shortest path between $v$ and $w$ in the graph induced by $X_{v}$. In particular, if $v=\min\mathrm{WReach}_{r}[G,L,w]$, then the path $P_{v,w}$ is a shortest path between $v$ and $w$ in $G$. *

Proof.

The pseudocode is given in Algorithm 4. First, using theorem 3, we compute for a given graph $G$ an order $L$ witnessing that $\mathrm{wcol}_{2r}(G)\leq c(2r)$ in $\mathcal{O}(r^{2}\log n)$ communication rounds.

The procedure implicitly uses an algorithm of Barenboim and Elkin [11], which assigns to each vertex $v$ a class id $cl(v)$, which together with the unique vertex identifier induces the linear order $L$. For ease of presentation, we write $v_{i}$ for the vertex at position $i$ in the order $L$ and call $i$ the super-id of the vertex.

The remaining part of the computation has $2r$ rounds which correspond to $2r$ rounds of a breadth-first search as in Algorithm 3. This time, the search is performed in parallel and we have to make sure that only a logarithmic amount of information is sent by every vertex for the $\mathcal{CONGEST}_{\textsc{BC}}$ model. The idea is that every vertex $w$ forwards only information about paths that start in a vertex $v\in\mathrm{WReach}_{2r}[G,L,w]$.

Every vertex $w$ maintains a set $\mathcal{P}_{w}$ of paths of length at most $2r$ from vertices $v\in\mathrm{WReach}_{2r}[G,L,w]$. For every vertex $v\neq w$ there is at most one path $P_{v}$ in $\mathcal{P}_{w}$ that starts in $v$ and certifies that $v\in\mathrm{WReach}_{2r}[G,L,w]$. In the first round, every vertex broadcasts its super-id, which we understand as a path of length [math]. A vertex $w$ receives super-ids and stores only those which are smaller than its own super-id.

In a later iteration, every vertex $w$ receives some sets of paths from its neighbors and computes their union $\mathcal{P}$. For every vertex $u_{1}$, with a super-id greater than the super-id of $w$, all paths from $\mathcal{P}$ starting in $u_{1}$ are discarded. For every vertex $u_{1}$ with a smaller super-id, vertex $w$ selects the shortest path starting in $u_{1}$ among all paths in $\mathcal{P}$ and $\mathcal{P}_{w}$. (There is at most one path in $\mathcal{P}_{w}$ that starts in $u_{1}$.) If there are many such shortest paths, $w$ chooses the lexicographically least one (with respect to the super-ids). Let this path be $P=u_{1},\ldots,u_{j}$ for some $j\leq 2r$. Then $P$ is stored in $\mathcal{P}_{w}$ (if there is already a path in $\mathcal{P}_{w}$ that starts in $u_{1}$, it is replaced by $P$). If $j<2r$, then $w$ broadcasts the path $u_{1},\ldots,u_{j},w$.

Observe that every vertex $w$ forwards information about a vertex $v$ only if $v\in\mathrm{WReach}_{2r}[G,L,w]$. Hence, $w$ forwards only at most $c$ paths simultaneously and the whole procedure works in the $\mathcal{CONGEST}_{\textsc{BC}}$ model. Observe also that we perform a breadth-first search through the cluster $X_{v}$ and break ties according to the order by vertex super-ids. This implies our claims on shortest paths. $\square$

We can now combine theorem 4 and lemma 7 to obtain the first main theorem of this section.

Theorem 8**.**

*Let $\mathcal{C}$ be a class of bounded expansion. There is a distributed algorithm which for every graph $G\in\mathcal{C}$ and every $r\in\mathbb{N}$ computes a representation of a sparse $r$-neighborhood cover in the $\mathcal{CONGEST}_{\textsc{BC}}$ model in $\mathcal{O}(r^{2}\cdot\log n)$ communication rounds. More precisely, the algorithm computes an order $L$, represented by $\log n$-sized labels and for every vertex $v$ a routing scheme of length at most $2r$ to every vertex in $\mathrm{WReach}_{2r}[G,L,v]$. *

Also theorem 5 can now be implemented as a distributed algorithm.

Theorem 9**.**

*Let $\mathcal{C}$ be a class of bounded expansion and let $r\in\mathbb{N}$. There is a constant $c(r)$ and a distributed algorithm which for every graph $G\in\mathcal{C}$ computes a $c(r)$-approximation of a minimum distance-$r$ dominating set in the $\mathcal{CONGEST}_{\textsc{BC}}$ model in $\mathcal{O}(r^{2}\cdot\log n)$ communication rounds. *

Proof.

Recall that we want to compute the distance-$r$ dominating set

[TABLE]

that is, every vertex $w$ elects the smallest vertex from its $r$-neighborhood with respect to $L$ to the distance-$r$ dominating set. As $w$ knows $\mathrm{WReach}_{r}[G,L,w]$ and a routing scheme to these vertices, all vertices can send to the smallest vertex in the list a short message that it should be included in the dominating set. Observe that if a vertex $u$ has to forward the identifier of a vertex $w\in\mathrm{WReach}_{r}[G,L,v]$ from some other vertex $v$, then also $w\in\mathrm{WReach}_{r}[G,L,u]$. Hence, no vertex has to forward more than $c(r)$ messages of total size at most $\mathcal{O}(c(2r)^{2}\cdot r\cdot\log n)$: every vertex $v$ forwards id’s of at most $c(2r)$ vertices $w$ from its $\mathrm{WReach}_{2r}[G,L,v]$ together with their rooting schemes (of size at most $2r$) to at most $c(2r)$ vertices in $\mathrm{WReach}_{2r}[G,L,w]$. $\square$

5 Connected Dominating Sets

In this section we study the Connected Distance-$r$ Dominating Set problem. Our main result in this section is the following theorem.

Theorem 10**.**

*Let $\mathcal{C}$ be a class of bounded expansion and let $r\in\mathbb{N}$. There is a constant $c$ and a distributed algorithm which for every graph $G\in\mathcal{C}$ computes a $c$-approximation of a minimum connected distance-$r$ dominating set in the $\mathcal{CONGEST}_{\textsc{BC}}$ model in $\mathcal{O}(r^{2}\cdot\log n)$ communication rounds. *

The following observation is folklore.

Lemma 11**.**

*Let $G$ be a connected graph and let $D$ be a distance-$r$ dominating set of $G$. Let $\mathcal{P}$ be a set of paths in $G$ such that for each pair $u,v\in D$ with $\mathrm{dist}(u,v)\leq 2r+1$ there is a path $P_{u,v}\in\mathcal{P}$ connecting $u$ and $v$. Then the subgraph $H$ induced by $D\cup\bigcup_{P\in\mathcal{P}}V(P)$ is connected. *

Proof.

We show by induction on $\mathrm{dist}(u,v)$ that all $u,v\in D$ are connected in $H$. The claim holds by definition of $H$ if $\mathrm{dist}(u,v)\leq 2r+1$. Now assume that $\mathrm{dist}(u,v)\geq 2r+2$ and let $P=(u=v_{0},v_{1},\ldots,v_{t}=v)$ with $t\geq 2r+3$ be a shortest path connecting $u$ and $v$. As $P$ is a shortest path, neither $u$ nor $v$ dominate $v_{r+1}$. Hence there is another vertex $w\in D$ which dominates $v_{r+1}$. As $v_{1}$ and $v_{r+1}$ are connected by a path of length $r$, and $w$ and $v_{r+1}$ are connected by a path $(w=w_{0},w_{1},\ldots,w_{r^{\prime}}=v_{r+1})$ of length $r^{\prime}\leq r$, $\mathrm{dist}(u,w)\leq 2r$, hence $w$ and $u$ are connected in $H$. Furthermore, the path $P^{\prime}=(w,w_{1},\ldots,w_{r^{\prime}-1},v_{r+1},\ldots,v_{t}=v)$ is shorter than $P$, hence, by induction hypothesis, $w$ and $v$ are connected in $H$. This implies that $u,v$ are connected in $H$. $\square$

Now, we use the local separation properties of the weak colouring numbers to connect the dominating set we computed in theorem 9. The proof of the following lemma is immediate by definition of weak reachability.

Lemma 12**.**

*Let $G$ be a graph and let $L$ be a linear order on $V(G)$. Let $u,v\in V(G)$ be such that there exists a path $P$ between $u$ and $v$ of length at most $r$. Let $w$ be the minimal vertex of $P$ with respect to $L$. Then $w\in\mathrm{WReach}_{r}[G,L,u]$ and $w\in\mathrm{WReach}_{r}[G,L,v]$. *

Corollary 13**.**

*Let $G$ be a connected graph and let $L$ be a linear order on $V(G)$. Let $D$ be an $r$-dominating set of $G$. Let $D^{\prime}$ be a set which is obtained by adding for each $v\in D$ and each $w\in\mathrm{WReach}_{2r+1}[G,L,v]$ the vertex set of a path between $v$ and $w$. Then $D^{\prime}$ is a connected distance-$r$ dominating set of $G$. *

Proof.

Fix a set $\mathcal{P}$ of paths in $G$ such that for each pair $u,v\in D$ with $\mathrm{dist}(u,v)\leq 2r+1$ there is a path $P_{u,v}\in\mathcal{P}$ connecting $u$ and $v$. According to lemma 11, the subgraph $H$ induced by $D\cup\bigcup_{P\in\mathcal{P}}V(P)$ is connected. According to lemma 12, for each path $P_{u,v}$ between $u$ and $v$, there is a vertex $w\in V(P_{u,v})$ is weakly $2r+1$-reachable both from $u$ and from $v$. As $D^{\prime}$ contains the vertex set of a path between $u$ and $w$ and of a path between $v$ and $w$, it follows that $D^{\prime}$ is a connected distance-$r$ dominating set of $G$. $\square$

We are now ready to prove the main theorem.

Proof. (Proof of theorem 10)

Instead of computing an order $L$ for $\mathrm{wcol}_{2r}(G)$ as in theorem 9, we compute an order $L$ for $\mathrm{wcol}_{2r+1}(G)$. Assume $|\mathrm{WReach}_{2r+1}[G,L,v]|\leq c^{\prime}$ for all $v\in V(G)$. We compute an $r$-dominating set $D$ based on the order $L$. Note that in Section 4 we used $L$ computed for parameter $2r$ and now we use $L$ computed for $2r+1$, but for all orders $L$ and all $v\in V(G)$ we have $\lvert\mathrm{WReach}_{2r}[G,L,v]\rvert\leq\lvert\mathrm{WReach}_{2r+1}[G,L,v]\rvert$.

By theorem 5, the set $D$ is at most $c^{\prime}$ times larger than a minimum distance-$r$ dominating set. As a by-product, see lemma 7, every vertex $v$ learns a path of length at most $2r+1$ to each $w\in\mathrm{WReach}_{2r+1}[G,L,v]$. Now, every vertex broadcasts its set of paths to construct the set $D^{\prime}$. As in the proof of theorem 9, observe that if a vertex $x$ has to forward a path from $w\in\mathrm{WReach}_{r}[G,L,v]$ to $v$ for some other vertex $v$, then also $w\in\mathrm{WReach}_{r}[G,L,x]$. Hence, no vertex has to forward more than $c^{\prime}$ messages of total size at most $\mathcal{O}(c^{\prime}\cdot r\cdot\log n)$. Clearly, the computed set $D^{\prime}$ has size at most $c^{\prime}\cdot(2r+1)\cdot|D|$ and by corollary 13 it is a connected distance-$r$ dominating set. We conclude be defining $c\coloneqq c^{\prime 2}\cdot(2r+1)$. $\square$

We now show how to use the greater power of the $\mathcal{LOCAL}$ model to compute connected dominating sets with much smaller constants involved. Our theorem is based on the simple observation that in the $\mathcal{LOCAL}$ model we can construct for every connected graph from an $r$-dominating set $D$ a connected depth-$r$ minor with $|D|$ vertices. This minor (by definition of bounded expansion classes) has only a linear number of edges and we can hence choose a set of short paths realizing the corresponding connections to connect the dominating set.

We want to define a partition of $V(G)$ into balls around vertices from an r-dominating set $D$. For a connected graph $G$ and an injection $id\colon V(G)\to\mathbb{N}$, we define the lexicographic order $<_{\mathrm{lex}}$ on the set of paths in $V(G)$ with respect to $id$ as follows. Consider two paths $P_{1}=v_{1},\ldots,v_{k}$ and $P_{2}=w_{1},\ldots,w_{\ell}$. If $k<\ell$, then $P\leq_{\mathrm{lex}}P_{2}$. If $k=\ell$, then $P_{1}\leq_{\mathrm{lex}}P_{2}$ if the sequence $id(v_{1}),\ldots,id(v_{k})$ is lexicographically smaller than the sequence $id(w_{1}),\ldots,id(w_{\ell})$ or $P_{1}=P_{2}$. For vertices $v,w\in V(G)$, let $P(v,w)$ be the lexicographically shortest path from $v$ to $w$.

Let $G$ be a connected graph, let $id(v)$ be the unique identifier of $v$ and let $D$ be a distance-$r$ dominating set of $G$. For each $v\in D$ let

[TABLE]

The $D$-partition $\mathcal{B}(D)$ of $G$ with respect to $id$ is the set \{B(v)\leavevmode\nobreak\ :\leavevmode\nobreak\ $v\in D\}$.

Lemma 14**.**

*Let $G$ be a connected graph and let $D$ be a distance-$r$ dominating set of $G$. Then $\mathcal{B}(D)=\{B(v):v\in D\}$ is a partition of $V(G)$ and $G[B(v)]$ has radius at most $r$ for all $v\in D$. *

Proof.

As $G$ is connected and $D$ is a distance-$r$ dominating set, $\mathcal{B}$ is a partition of $V(G)$. Furthermore, for each $w\in V(v)$, there is a lexicographically shortest path $P$ of length at most $r$ from $v$ to $w$ in $G$. Assume towards a contradiction that $P$ is not also a path in $B(v)$. Then there is $z\in V(P)$ and $u\in D$ such that $z\in B(u)$. By definition of $B(u)$, the lexicographically shortest path $Q^{\prime}$ from $u$ to $z$ is smaller than the lexicographically shortest path $Q$ between $v$ and $z$. But then the path $P^{\prime}$ obtained by replacing the initial part $Q$ of $P$ by $Q^{\prime}$ is lexicographically smaller than $P$, a contradiction. $\square$

Lemma 15**.**

*Let $G$ be a connected graph and let $D$ be a distance-$r$ dominating set of $G$. By contracting the sets $B(v)$ for $v\in D$, we obtain a connected depth-$r$ minor of $G$. *

Proof.

It is immediate by definition of depth-$r$ minors and lemma 14 that we construct a depth-$r$ minor $H\preccurlyeq_{r}G$. Furthermore, as $G$ is connected and as $\mathcal{B}(D)$ is a partition of $V(G)$ by the same lemma, it is easy to see that $H$ is connected. $\square$

Lemma 16**.**

*Let $G$ be a connected graph such that for each depth-$r$ minor $H\preccurlyeq_{r}G$ we have $\lvert E(H)\rvert\leq d\cdot\lvert V(H)\rvert$. Let $D$ be a distance-$r$ dominating set of $G$. We can compute a connected dominating set $D^{\prime}$ of $G$ of size at most $2r\cdot d\cdot\lvert D\rvert$ in $3r+1$ communication rounds in the $\mathcal{LOCAL}$ model. *

Proof.

In this proof, we write $H(D)$ for the depth-$r$ minor constructed from a distance-$r$ dominating set $D$ as in lemma 15.

Every vertex $v\in D$ can find its $2r+1$-neighborhood in $2r+1$ communication rounds. With this information, each $v\in D$ can construct $B(v)$, as all possible dominators for $w\in N_{r}[v]$ must come from $N_{2r}[v]$. Each vertex $v\in D$ (now understood as representing a vertex of $H(D)$) can also learn its neighbors in $H(D)$ (here we need to learn $N_{2r+1}[v]$). Now each vertex $v$ computes the lexicographically shortest path $P_{uv}$ of length at most $2r+1$ for each neighbor $u$ in $H(D)$ (take the ordering induced by vertex id’s). Observe that $u$ and $v$ fix the same path $P_{vu}$, hence, the two vertices can report to all vertices on $P_{vu}$ in another $r$ communication rounds that they shall be included in the connected dominating set $D^{\prime}$.

By lemma 15, the constructed set $D^{\prime}$ is a connected distance-$r$ dominating set. Furthermore, by assumption, $H(D)$ has at most $d\cdot|D|$ many edges. Each edge is replaced by at most $2r-1$ vertices in the above construction. Adding the $|D|$ vertices of the original set $D$, we obtain the claimed bounds. $\square$

As a corollary from lemma 14 and lemma 16 we obtain the following theorem.

Theorem 17**.**

*Let $\mathcal{C}$ be a class of graphs of bounded expansion and assume that for every graph $G\in\mathcal{C}$ we can compute a $c$-approximation $D$ of a minimum distance-$r$ dominating set of $G$ in $t$ rounds in the $\mathcal{LOCAL}$ model. Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ denote the edge density function of depth-$r$ minors of $\mathcal{C}$. Then there is a distributed algorithm which finds a $2rcf(r)$-approximation for connected distance-$r$ dominating set of $G$ in $\mathcal{O}(t+r)$ rounds in the $\mathcal{LOCAL}$ model. *

The theorem can be applied, e.g., to extend the algorithm of Lenzen et al. [36] to obtain a connected dominating set on planar graphs in the local model which is only $6$ times larger than the dominating set computed for the planar graph (an $n$-vertex planar graph has at most $3n-6$ edges). Similarly, it applies to the extension of Lenzen et al.’s algorithm by Amiri et al. [5] for graphs of bounded genus or to the randomized $\mathcal{O}(a^{2})$ approximation of Lenzen and Wattenhofer [38] applied to graphs with excluded minors (here, $a\in\mathcal{O}(t\log t)$ if $K_{t}$ is excluded as a minor).

6 Conclusion

What are the most general classes of graphs that admit efficient algorithms for certain problems? The ambitious goal to answer this question for the dominating set problem has lead to strong graph theoretic and algorithmic results once it was known that it cannot be solved efficiently in full generality. Lower bounds both in classical complexity and in distributed complexity have motivated the study of more and more general graph classes.

Bounded expansion classes of graphs are very general classes of sparse graphs. In this paper we proposed a new constant factor approximation algorithm for the (Connected) Distance-$r$ Dominating Set problem on these classes of graphs. The algorithm improves the previously best known approximation algorithm by Dvořák [21], however, its main feature is that it is tailored to be implemented in a distributed setting.

It was proved in [10] that on the class of graphs with arboricity at most $a$ the size of a minimum dominating set can be approximated by a factor $3a$ by an LP rounding algorithm, but that it is NP-hard to approximate the size of the minimum dominating set to within $a-1-\varepsilon$ in this class for every $\varepsilon>0$. This natural leads to consider that bounded average degree (or bounded arboricity) is a natural requirement for a class of graphs closed under taking subgraphs to allow constant factor approximation for the size of the minimum dominating set. By considering subdivisions, it follows that the property of every graph present as an $r$-subdivision in the class should have average degree at most $C(r)$ — that is of having bounded expansion — is a natural requirement to allow constant factor approximation for the size of the distance-$r$ minimum dominating sets for every $r$.

Our techniques are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes of graphs. Formerly, no distributed algorithms that compute such covers were known and we believe that these techniques are interesting beyond the presented applications of computing (connected) dominating sets. We pose the question whether sparse neighborhood covers can be computed in distributed constant time. This question is open even on more restrictive graph classes, e.g., on planar graphs, where dominating sets can be approximated in constant time.