    # An Õ(log^2 n)-approximation algorithm for 2-edge-connected dominating set

In the Connected Dominating Set problem we are given a graph G=(V,E) and seek a minimum size dominating set S ⊆ V such that the subgraph G[S] of G induced by S is connected. In the 2-Edge-Connected Dominating Set problem G[S] should be 2-edge-connected. We give the first non-trivial approximation algorithm for this problem, with expected approximation ratio Õ(log^2n).

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## 1 Introduction

Let be a graph. A subset of nodes of is a dominating set in if every has a neighbors in . In the Dominating Set problem the goal is to find a min-size dominating set . In the Connected Dominating Set problem the subgraph of induced by should be connected. This problem admits a tight approximation ratio , even in the node weighted case [10, 11], based on .

A graph is -edge-connected if it contains edge disjoint paths between every pair of nodes. We consider the following problem.

-Edge-Connected Dominating Set Input:   A graph . Output: A min-size dominating set such that is -edge-connected.

Given a distribution over spanning trees of a graph , the stretch of is , where denotes the distance between and in a graph . Let denote the lowest known upper bound on the stretch that can be achieved by a polynomial time construction of such for a graph on nodes. By the work of Abraham, Bartal, and Neiman , that is in turn based on the work of Elkin, Emek, Spielman, and Teng , we have:

 σ(n)=O(logn⋅loglogn⋅(logloglogn)3)=~O(logn) .

Our main result is:

###### Theorem 1.1

-Edge-Connected Dominating Set admits an approximation algorithm with expected approximation ratio .

In the rest of the Introduction we discuss motivation, related problems, and give a road-map of the proof of Theorem 1.1.

It is a common problem in network design to route messages through the network. Many routing protocols exploit flooding strategy in which every node broadcasts the message to all of its neighbors. However, such protocols suffer from a large amount of redundancy. Ephremides, Wieselthier, and Baker  introduced the idea of constructing a virtual backbone of a network. A virtual backbone is often chosen to be a connected dominating set – a connected subgraph (a tree) on a dominating node set. Then only the nodes of the tree are involved in the routing, which may significantly reduce the number of messages the routing protocol generates. Moreover, we only need to maintain the nodes in the tree to keep the message flow. This raises the natural problem of constructing a “cheap” connected virtual backbone . Usually “cheap” means that should have a minimum number of edges or nodes, or, more generally, that we are given edge costs/node weights, and should have a minimum cost/weight.

In many cases we also require from the virtual backbone to be robust to edge or node failures. A graph is -edge-connected if it contains edge disjoint paths between every pair of nodes; if the paths are required to be internally node disjoint then is -connected. A subset of nodes in a graph is an -dominating set if every has at least neighbors in . In the Min-Weight -Connected -Dominating Set problem we seek a minimum node weight -dominating set such that the subgraph of induced by is -connected. This problem was studied in many papers, both in general graphs and in unit disk graph, for arbitrary weights and also for unit weights; the unit weights case is the -Connected -Dominating Set problem. We refer the reader to recent papers [8, 21, 18]. In the Min-Cost -Connected -Dominating Subgraph problem, we seek to minimize the cost of the edges of the subgraph rather than the weight of the nodes. We observe that for unit weights/costs, the approximability of the -Connected -Dominating Set problem is equivalent to the one of the -Connected -Dominating Subgraph problem, up to a factor of ; this is so since the number of edges in a minimally -connected graph is between and . The same holds also for the -edge-connectivity variant of these problems.

Most of the work on the Min-Weight -Connected -Dominating Set problem focused on the easier case , when the union of a partial solution and a feasible solution is always feasible. This enables to construct the solution by computing first an -approximate -dominating set and then a -approximate augmenting set to satisfy the connectivity requirements; the approximation ratio is then bounded by the sum of the ratios of the two sub-problems. The currently best ratios when are : for general graphs, for unit disc graphs, and for unit disc graphs with unit node weights. However, when this approach does not work, and the only non trivial ratio known is for (unweighted) unit disk graphs, due to Wang et al. , where they obtained a constant ratio for and . It is an open question to obtain a non-trivial ratio for the (unweighted) -Connected Dominating Set problem in general graphs.

The -Edge-Connected Dominating Set problem that we consider is the edge-connectivity version of the above problem, when the virtual backbone should be robust to single edge failures. As was mentioned, the approximability of this problem is the same, up to a factor of , as that of the -Edge-Connected Dominating Subgraph problem that seeks to minimize the number of edges of the subgraph rather than the number of nodes. We prove Theorem 1.1 for the latter problem using a two stage reduction. Our overall approximation ratio is a product of the first reduction fee and the approximation ratio for the problem obtained from the second reduction.

In the first stage (see Section 2) we use the probabilistic embedding into a spanning tree of  with stretch to reduce the problem to a “domination version” of the so called Tree Augmentation problem (c.f. ); in our problem, which we call Dominating Subtree, we are given a spanning tree in and seek a min-size edge set and a subtree of , such that dominates all nodes in and is -edge-connected. This reduction invokes a factor of in the approximation ratio. Gupta, Krishnaswamy, and Ravi  used such tree embedding to give a generic framework for approximating various restricted -edge-connected network design problems, among them the -Edge-Connected Group Steiner Tree problem. However, all their algorithms are based on rounding an appropriate LP relaxations, while our algorithm is purely combinatorial and uses different methods.

In the second stage (see Section 3) we reduce the Dominating Subtree problem to the Subset Steiner Connected Dominating Set problem . While we show in Section 4 that in general this problem is as hard as the Group Steiner Tree problem, the instances that are derived from the reduction have special properties that will enable us to obtain ratio . We note that the reduction we use is related to the one of Basavaraju et al. , that showed a relation between the Tree Augmentation and the Steiner Tree problems.

## 2 Reduction to the dominating subtree problem

To prove Theorem 1.1 we will consider the following variant of our problem:

-Edge-Connected Dominating Subgraph Input:   A graph . Output: A -edge-connected subgraph of with minimum such that is a dominating set in .

Since holds for any edge-minimal -edge-connected graph , then if -Edge-Connected Dominating Subgraph admits ratio then -Edge-Connected Dominating Set admits ratio . Thus it is sufficient to prove Theorem 1.1 for the -Edge-Connected Dominating Subgraph problem.

For simplicity of exposition we will assume that we are given a single spanning tree with stretch , namely that

 |Tf|≤σ    ∀f∈E∖ET

where denotes the path in the tree between the endnodes of . We say that covers if . For an edge set let denote the forest formed by the tree edges of that are covered by the edges of . The following two lemmas give some cases when is a -edge-connected graph.

###### Lemma 1

If then is -edge-connected if and only if is a tree.

###### Proof

Note that every has both ends in . It is known (c.f. ) that if is a tree and is an additional edge set on the node set of , then is -edge-connected if and only if . This implies that if is a tree then is -edge-connected. Now suppose that is not a tree. Let be a connected component of . Then no edge in has exactly one end in . Thus is also a connected component of , so is not connected. ∎

###### Lemma 2

If is a -edge-connected subgraph of then is a tree.

###### Proof

By Lemma 1 the statement is equivalent to claiming that is -edge-connected. To see this, note that is obtained from the -edge-connected graph by sequentially adding for each the path . It is known that adding a simple path between two nodes of a -edge-connected graph results in a -edge-connected; this is so also if contains some edges of the graph. The statement now follows by induction. ∎

Let us consider the following problem.

Dominating Subtree Input:   A graph and a spanning tree in . Output: A min-size edge set such that is a dominating tree.

From Lemmas 1 and 2 we have the following.

###### Corollary 1

Let be an optimal solution of a -Edge-Connected Dominating Subgraph instance . Let be a spanning tree in with stretch and a -approximate solution to the Dominating Subtree instance . Then is a feasible solution to the -Edge-Connected Dominating Subgraph instance and .

###### Proof

is a feasible solution to the -Edge-Connected Dominating Subgraph instance by the definition of the Dominating Subtree problem and Lemma 1. By Lemma 2, is a feasible solution to the Dominating Subtree instance, thus . Since has stretch we get . ∎

Hence to finish the proof of Theorem 1.1 is is sufficient to prove the following theorem, that may be of independent interest.

###### Theorem 2.1

The Dominating Subtree problem admits approximation ratio .

## 3 Reduction to subset connected dominating set

In this section we reduce the Dominating Subtree problem to the Subset Steiner Connected Dominating Set, and show that the special instances that arise from the reduction admit ratio . The justification of the reduction is given in the following lemma.

###### Lemma 3

Let be a tree and and edge set on , and let . Let be a bipartite graph where . Then has edge disjoint -paths if and only if has an -path.

###### Proof

Let be the -path in . By Menger’s Theorem, has edge disjoint -paths if and only if for every there is that covers . Let . Let be the set of edges in uncovered by . We need to show that if and only if .

Suppose that . Among the nodes in , let be the furthest from along . Let be the node in after . Then , since . We claim that . Otherwise, there is with and we get that (since ), contradicting the choice of .

Suppose that there is . Let be the two trees of , where and . Since no link in covers , every link in has both ends either in or in . This implies that no node in belongs to . ∎

Let us assume w.l.o.g. that our Dominating Subtree instance consists of a tree on and an edge set on that contains no edge from .

###### Definition 1

Given a Dominating Subtree instance the connectivity-domination graph has node set and edge set where (see Fig. 1):

 I = {fg:f,g∈E,V(Tf)∩V(Tg)≠∅} D = {ev:e∈E,v∈V,v∈Te or v is a neighbor of % Te in G} Figure 1: Illustration to Definition 1. Here E={f1,f2,f3,f4} and {f1,f3} is a unique optimal solution. In the connectivity-domination graph on the right, the edges in I are shown by bold arcs, and the edges in D are shown by straight lines; the “membership edges” shown by bold lines encode that v∈Tf, while the other edges in D shown by thin lines encode that v is dominated but does not belong to Tf.

Note that an edge in encodes that and have a node in common, while an edge encodes that is dominated by (belongs to or is connected by an edge of to some node in ). From Lemma 3 we have:

###### Corollary 2

is a feasible solution to a Dominating Subtree instance if and only if in the connectivity-domination graph the following holds:
(i) is connected; (ii) dominates .

Our goal is to give an approximation algorithm for the problem of finding min-size as in Corollary 2. Note that in this problem are both subsets of nodes of . This is a particular case of the following problem.

Subset Steiner Connected Dominating Set Input:   A graph and a partition of . Output: A min-size such that is connected and dominates .

In Section 4 we observe that up to constants, the approximability of this problem is the same as that of the Group Steiner Tree problem, that admits ratio . However, in some cases better ratios are possible. In the case we get the (unweighted) Steiner Connected Dominating Set problem, that admits ratio . We show ratio when is the connectivity-domination graph, with and . In what follows, given a Subset Steiner Connected Dominating Set instance, let be the least integer such that for every , any two neighbors of in are connected by a path in that has at most internal nodes.

###### Lemma 4

Subset Steiner Connected Dominating Set admits approximation ratio if partition and is an independent set in .

###### Proof

We find an approximate solution for the Steiner Connected Dominating Set instance (with ) using the algorithm of . If then we are done. Else, let be a subtree of with node set . We may assume that has no leaf in , otherwise such leaf can be removed from and from . Let and . Since is an independent set in , dominates , and the nodes in are used in just to connect between the nodes in . Moreover, since is an independent set in

 ∑r∈RT(degT(r)−1)≤|ET|−|RT|=|S|−1−|RT|=|QT|−1 .

Let . Add a set of dummy edges that form a tree on the neighbors of in , and then replace every dummy edge by a path in that has at most internal nodes. Applying this on every gives a connected graph in that contains the set that dominates , and the number of nodes in this graph is at most

 |QT|+q∑r∈RT(degT(r)−1)≤(q+1)(|QT|−1)≤(q+1)|S| .

Since is times the optimum, the lemma follows. ∎

Note that in our case, when is the connectivity-domination graph, we have and . Then partition and is an independent set in , by the construction. The next lemma shows that is a small constant in our case.

###### Lemma 5

if is the connectivity-domination graph and , .

###### Proof

Let and let be the set of neighbors of in . For let be defined as follows:

• If there exists some (possibly ) such that and have a node in common, then we say that is of type 1 and let .

• If as above does not exist then we say that is of type 2 and let .

For illustration, consider the example in Fig. 1.

• Let . Then . If or if then , hence is of type 1 and we may have . If then , but is still of type 1 since for we have and have a node in common.

• Let . Then . There is no such that , hence both are of type 2.

Suppose that every is of type 1. Then for every we have or , and note that . Thus for any , the sequence forms a path in with at most two internal nodes.

Suppose that there is of type 2. Then , and is dominated by via some edge of ; otherwise is of type 1. Let and be the two subtrees of , where and . Note that no edge in connects and ; otherwise is of type 1. Hence is a bridge of . This implies consists of a single node , as otherwise the instance has no feasible solution. Consequently, every is of type 2 and holds, hence is a clique, and the lemme follows. ∎

This concludes the proof of Theorem 2.1, and thus also the proof of Theorem 1.1.

## 4 Connected Dominating Set variants

Here we make some observations about the approximability of several variants of the Connected Dominating Set (CDS) problem. In all these variants we are given a graph and possibly edge-costs/node-weights, and seek a minimum cost/weight/size subtree of that satisfies a certain domination property. Recall that in the CDS problem should dominate . The additional variants we consider are as follows.

Steiner CDS: dominates a given set of terminals .

Subset Steiner CDS: dominates and for a partition of .

Partial CDS: dominates at least nodes.

We relate these problems to the Group Steiner Tree (GST) problem: given a graph and a collection of groups (subsets) of , find a minimum edge-cost/node-weight/size subtree of that contains at least one node from every group. When the input graph is a tree and there are groups, edge-costs GST admits ratio , and this is essentially tight . For general graphs the edge-costs version admits ratio , using the result of  for tree inputs and the  probabilistic tree embedding. However, the best ratio known for the node-weighted GST is the one that is derived from the more general Directed Steiner Tree problem [3, 20, 14, 17] with terminals – for any integer , ratio in time .

As was observed in , several CDS variants are particular cases of the corresponding GST variants, where for every relevant node we have a group of nodes that dominate in the input graph. Specifically, we have the following.

###### Lemma 6

For edge-costs/node-weights, ratio for GST with nodes and groups implies ratio for Subset Steiner CDS, and this is so also for the unit node weights versions of the problems.

###### Proof

Given a Subset Steiner CDS instance with edge costs/node weights and , construct a GST instance by introducing for every a group of nodes in that dominate . In all cases, we return an -approximation for the GST instance on , that has nodes and groups. ∎

Earlier, Guha and Khuller  showed that the inverse is also true for edge-costs CDS and node-weighted Steiner CDS; in  the reduction was to the Set TSP problem, that can be shown to have the same approximability as GST, up to a factor of 2. Note that already the edge-costs CDS is GST hard, hence our ratio for unit edge costs -Edge-Connected Dominating Set is unlikely to be extended to arbitrary costs.

We now show that Subset Steiner CDS with unit edge costs/node weights is hard to approximate as GST with general edge costs. In what follows, we will assume that ratio for a given problem is an increasing function of .

###### Theorem 4.1

For any constant , ratio for Subset Steiner CDS with unit edge costs/node weights implies ratio for GST with arbitrary edge costs.

Theorem 4.1 is proved in the next two lemmas. Note that combined with Lemmas 6, Theorem 4.1 implies that the approximability of Subset Steiner CDS with unit edge costs/node weights is essentially the same as that of GST with arbitrary edge costs, up to a constant factor. Recall that we showed that particular instances of Subset Steiner CDS with unit node weights admit ratio . Theorem 4.1 implies that we could not achieve this for general unit node weights instances.

The next lemma shows that for unit edge costs/node weights, Subset Steiner CDS is not much easier than GST.

###### Lemma 7

For unit edge costs/node weights, ratio for Subset Steiner CDS implies ratio for GST with groups.

###### Proof

For each one of the problems in the lemma, any inclusion minimal solution is a tree, hence the unit edge costs case is equivalent to the unit node weights case; this is so up to an additive term, which can be avoided by guessing an edge/node that belongs to some optimal solution. So we will consider just the unit node weights case. Given a unit weight GST instance construct a unit weight Subset Steiner CDS instance as follows. For each group add a new node connected to all nodes in . The set of nodes that should be dominated is , and . Any subtree of is is also a subtree of , and for any group , contains a node from if and only if dominates . Thus is a feasible GST solution if and only if is a feasible Subset Steiner CDS solution. ∎

The next lemma shows that GST with unit edge costs is not much easier than GST with arbitrary edge costs.

###### Lemma 8

If GST with unit edge costs admits ratio then for any constant GST (with arbitrary edge costs) admits ratio .

###### Proof

Let be a GST instance (with arbitrary edge costs). Fix some optimal solution . Let be the maximum cost of an edge in . Note that . Since there are edges, we can guess , and remove from all edges of cost greater than .

Let . Define new costs by . Note that for all , thus for any edge set with

 c(J)=∑e∈Jc(e)≤∑e∈Jμ(c′(e)+1)=μc′(J)+μ|J|≤μc′(J)+ϵM .

This implies that if is an -approximate solution w.r.t. the new costs then

 c(H)≤μc′(H)+ϵM≤μαc′(H∗)+ϵM≤αc(H∗)+ϵc(H∗)=(α+ϵ)c(H∗) .

So ratio w.r.t. costs implies ratio w.r.t. the original costs .

The instance with costs can be transformed into an equivalent instance with unit edge costs and at most nodes by a folklore reduction that replaces every edge by a path of length equal to the cost of the edge. Note that is integer valued and that

 c′(e)≤c(e)μ=c(e)⋅nϵM≤nϵ .

Thus the number of nodes in the obtained unit edge costs instance is bounded by .

Given the GST instance with costs , we contract every zero cost edge while updating the groups accordingly. Then we replace every edge by a -path of length , thus obtaining an equivalent GST instance with unit edge costs and at most nodes. ∎

Theorem 4.1 follows from the last two lemmas.

Finally, we consider the Partial CDS problem. For unit node weights the problem was shown to admit a logarithmic ratio in . We show that in the case of arbitrary weights, the problem is not much easier than Subset Steiner CDS, and thus also is not easier than GST.

###### Lemma 9

For edge costs/node weights, ratio for Partial CDS implies ratio for Subset Steiner CDS.

###### Proof

Let us consider the case of node-weights. Given a Subset Steiner CDS instance construct a Partial CDS instance as follows. The graph is obtained from by adding copies of , and for each connecting each copy of to all nodes in that dominate . We let if and otherwise, and we let . In the obtained Partial CDS instance, a subset of that does not dominate , dominates at most nodes; hence any feasible solution of finite weight must dominate . The Partial CDS instance has nodes, and the node weights case follows.

In the case of edge costs are as in the case of node weights, and the cost of an edge of is if and otherwise. The rest of the proof is the same as in the case of node-weights. ∎

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