Noname manuscript No.
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On connected domination in unit ball graphs
Sergiy Butenko · Sera Kahruman-Anderoglu ·
Oleksii Ursulenko
Received: date / Accepted: date
Abstract Given a simple undirected graph, the minimum connected dominating set
problem is to find a minimum cardinality subset of vertices D inducing a connected
subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual
routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit
ball graphs and then introduces and studies the edge-weighted bottleneck connected
dominating set problem, which seeks a minimum edge weight in the graph such that
the corresponding bottleneck subgraph has a connected dominating set of size k. In
wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show
that the problem is hard to approximate within a factor better than 2 in graphs whose
edge weights satisfy the triangle inequality and provide a 3-approximation algorithm
for such graphs.
Keywords Connected dominating set · Unit ball graph · Wireless networks
1 Introduction
One of the traditional models of a wireless network topology is based on the concept
of a unit disk graph (UDG), in which the nodes represent points on the plane and the
edges connect the pairs of nodes corresponding to points located within a unit distance
from each other. The routing protocols in such networks are typically based on the
concept of a virtual backbone, which is a (small) subset of nodes that are used as a
core for communication within the network. In particular, connected dominating sets
This research was supported by AFOSR Award No. FA9550-08-1-0483
S. Butenko, S. Kahruman-Anderoglu, and O. Ursulenko
Department of Industrial and Systems Engineering, Texas A&M University
College Station, TX 77843-3131
Tel.: +1-979-458-1089
Fax: +1-979-847-9005
E-mail: {butenko,sera,ursul}@tamu.edu
2
(CDS) are often used to describe a virtual backbone in ad hoc wireless networks. This
paper studies two related issues: 1) computing small connected dominating sets in a
unit ball graph, representing a more realistic, three-dimensional, model of a wireless
network; 2) selecting an appropriate transmission range for wireless nodes allowing to
implement a CDS-based routing with a given size of CDS. We begin by providing the
necessary background and motivating the proposed research directions.
Consider a simple undirected graph G = (V, E) with the set of vertices V and the set
of edges E. For a subset W of vertices, we will denote by G[W ] the subgraph induced by
W , i.e., G[W ] = (W, E ∩ (W × W )). For a vertex v ∈ V , NG (v) = {u ∈ V : (u, v) ∈ E}
or simply N (v) denotes the neighborhood of v and degG (v) = |N (v)| is the degree of
∪
∪
∪
v. For S ⊆ V , let NG
(S) = {v ∈ V \ S|∃ u ∈ S : (u, v) ∈ E} and NG
[S] = NG
(S) ∪ S.
For a pair of vertices u, v, we denote by dG (u, v) the smallest number of edges in a
path connecting u and v in G. For a positive integer m, the m-th power of G is given
by Gm = (V, E m ), where E m = {(u, v) : dG (u, v) ≤ m}.
A subset I of vertices is called an independent set if G[I] has no edges. It is called
a maximal independent set if it is not a subset of a larger independent set. D ⊆ V is
called a dominating set of G if each vertex v ∈ V is either in D or has at least one
neighbor in D. For a connected graph G, a dominating set D is called a connected
dominating set (CDS) if it induces a connected subgraph G[D] in G. The cardinality of
a minimum connected dominating set (MCDS) in G is called the connected domination
number of G and is denoted by γc (G). Note that the problem of finding a minimum
connected dominating set in a graph is equivalent to the problem of finding a spanning
tree with maximum number of leaves in G, with all non-leaf nodes in such a spanning
tree forming a minimum connected dominating set.
A graph G is called a unit ball graph (UBG) if its vertices can be represented as
points in 3-dimensional Euclidean space <3 so that the distance between two points
corresponding to an edge in G is less than 1. Geometrically, this means that two
points u and v in <3 are connected by an edge if u is inside the ball or radius 1
centered in v. UBGs are gaining popularity in modeling ad hoc wireless networks,
where transmitting hosts are represented by dots in <3 and the unit length represents
the host’s transmission range (which is assumed to be the same for all hosts). Since the
UDGs can be viewed as a subclass of the UBGs, in which all dots are restricted to be
coplanar, most of the negative complexity results previously obtained for UDGs remain
valid for the class of UBGs. In particular, the maximum independent set, minimum
vertex cover, minimum dominating set, minimum independent dominating set, and
minimum connected dominating set problems are known to be NP-hard for UDGs [3],
and therefore are all NP-hard for UBGs as well. Moreover, given a graph without its
geometric representation, it is NP-hard to determine whether it can be represented as
a UDG or a UBG [1]. Furthermore, it is NP-hard to recognize a UDG even if it does
have a unit ball graph representation. The recognition of unit ball graphs restricted to
the case where balls can only touch one another is also proven to be NP-hard in [7].
There are several polynomial-time algorithms with performance guarantee for the
minimum connected dominating set problem. In particular, Guha and Khuller [6] propose an algorithm which gives a performance ratio of ln ∆+3, where ∆ is the maximum
degree of the graph. A Polynomial Time Approximation Scheme (PTAS) for the minimum connected dominating set problem in UDGs is also possible, as shown in [9]
and more recently in [2]. In a recent paper, Zhang et al. [17] propose a PTAS for the
minimum connected dominating set in UBGs. Furthermore, when their method, which
is based on shifting and partitioning, is applied on a UDG, the running time is im-
3
proved. The first step of this PTAS requires a ρ-approximation algorithm for minimum
connected dominating set problem in UBGs. The number of shifting operations is determined by the approximation parameter and ρ. Therefore, although there exists a
PTAS for this problem, designing approximation algorithms with good approximation
ratios is still important.
In a UDG (UBG) model of a wireless network, the unit of distance represents the
transmission range of a wireless node, which is assumed to be the same for each node.
There are several variations of these popular geometric graphs that are used to model
wireless network topology, such as double-disk graphs, quasi-unit disk graphs, unit ball
graphs in a doubling metric and bounded independence graphs. However, one common
feature of the traditional models is the assumption that the range of communication or
its estimate is given in advance, and the network topology, which is crucial in designing
routing protocols, is completely defined by the transmission range. Since for some types
of wireless networks, such as sensor networks, energy considerations are critical, this
assumption becomes especially important, and the choice of the transmission range
has to be approached with utmost care. Several papers investigate this problem with
the goal of ensuring connectivity [5, 15, 14, 11, 10, 12]. However, ideally, one would want
to tie the choice of transmission range to a specific routing protocol that will be used
for communication within the network. Motivated by the popularity of CDS-based
routing protocols, we propose to use the bottleneck connected dominating set problem
as a viable approach to selecting the transmission range of a wireless node in a network.
Given a complete graph G = (V, E) with positive edge weights (costs, distances) ce , e ∈
E, the bottleneck subgraph G(e) of G corresponding to the edge e is defined as follows:
G(e) = (V, E(e)), where E(e) = {e0 ∈ E : ce0 ≤ ce }.
We will call ce the cost of the bottleneck subgraph G(e). Given a subset of vertices
S in G, its bottleneck connected domination cost Cb (S) is defined as the minimum
cost of a bottleneck subgraph G(e) of G such that S forms a connected dominating
set in G(e). The k-bottleneck connected dominating set (k-BCDS) problem is to find
a subset of k vertices with minimum bottleneck connected domination cost in G. A
restricted version of k-BCDS problem, in which the edge weights are required to satisfy
the triangle inequality, will be denoted by k-BCDS(∆).
The k-BCDS problem considered in a UDG or a UBG representing a wireless
network optimizes the transmission range by ensuring a connected dominating set of
the predetermined size k. Several bottleneck problems were considered in the literature,
such as minimum bottleneck spanning tree, bottleneck traveling salesman problem,
etc. The bottleneck version of the dominating set problem that was studied previously
focuses on vertex-weighted graphs, where bottleneck cost is defined in terms of the
vertex weights, and can be solved in linear time [16, 13]. In our problem, we define the
bottleneck cost in terms of the edge weights. To the best of our knowledge, there are
no published results on the edge-weighted bottleneck dominating set problems.
The remainder of this paper is organized as follows. Section 2 establishes a relation
between the sizes of a maximal independent set and a minimum connected dominating
set in a unit-ball graph. The constant-ratio approximation algorithm for the minimum
connected dominating set problem and its performance analysis are presented in Section
3. Section 4 introduces and studies the edge-weighted bottleneck connected dominating
set problem in general and geometric graphs. Finally, Section 5 concludes the paper.
4
2 Preliminaries
Consider an equivalent geometric representation of a UBG, in which instead of the
balls of radius 1 we use the balls of diameter 1 centered in each point representing a
vertex of G. Then (u, v) ∈ E if and only if the open balls of diameter 1 centered in u
and v have a nonempty intersection (here and henceforth we will use the same notation
u for a point in <3 representing the vertex u of G).
Lemma 1 Given an arbitrary vertex v of a UBG G, its neighborhood N (v) cannot
contain an independent set of size greater than 12.
Proof We will use the geometric concept of kissing number, which is the maximum
number of spheres of radius 1 that can simultaneously touch the unit sphere in ndimensional Euclidean space. For n = 3, the kissing number is known to be equal to
12 [4], which implies the lemma’s statement.
Lemma 2 The size of any maximal independent set I of a UBG G is at most 11γc (G)+
1. Moreover, if max{|N (v) ∩ I| : v ∈ V \ I} ≤ ν ≤ 11 then |I| ≤ νγc (G).
Proof Let I = {u1 , . . . , up } be an arbitrary maximal independent set and let D =
{v1 , . . . , vq }, where q = γc (G), be a minimum connected dominating set in G. Without
loss of generality, we can assume that the vertices in D are listed in the order of an
arbitrary depth-first search performed on G[D]. Let us partition the vertices in I into
disjoint subsets I1 , . . . , Iq so that each vertex in Ij , j = 1, . . . , q, is a neighbor of vj ,
but is not a neighbor of any of the preceding vertices v1 , . . . , vj−1 in D. Then due to
Lemma 1, |I1 | ≤ 12. Note that for j = 2, . . . , q, at least one of the vertices v1 , . . . , vj−1
is adjacent to vj . Let us denote such a vertex by vj 0 . Again, |Ij | ≤ 12, and if we
assume |Ij | = 12, then at least one of vertices (say, uj 00 ) from Ij would have to be
a neighbor of vj 0 , which would mean that uj 00 is already included in one of the sets
I1 , . . . , Ij−1 . Therefore, |Ij | ≤ 11 for j = 2, . . . , q and we obtain |I| =
γcP
(G)
|Ij | ≤
j=1
12 + 11(γc (G) − 1) = 11γc (G) + 1. Similarly, if max{|N (v) ∩ I| : v ∈ V \ I} ≤ ν we
have |I| =
γcP
(G)
|Ij | ≤ νγc (G).
j=1
3 A 22-approximate algorithm for MCDS in UBG
In this section, we propose an approximation algorithm for the minimum connected
dominating set problem in a connected UBG. The idea is to build a maximal independent set (which is also a dominating set) I, and in the process connect it through
intermediate “parent” vertices that will guarantee connectivity. The algorithm consists
of three stages: initialization, construction, and improvement, proceeding as follows.
Initialization. This stage initializes the algorithm’s parameters. We use the following
notations: I denotes the constructed maximal independent set; W denotes the set of
∪
vertices adjacent to at least one vertex in I (i.e., W = NG
(I)); U is the set of vertices
that are neither in I nor in the neighborhood of I (we will call the vertices in U
unexplored vertices); and D denotes the connected dominating set being built. We pick
5
Algorithm 1: Approximating the minimum connected dominating set problem.
input a nontrivial, simple, undirected, connected graph G = (V, E);
/* Initialization */
pick v ∗ ∈ arg max{deg(v) : v ∈ V };
I ← {v ∗ };
D ← {v ∗ };
W ← N (v ∗ );
U ← V \ (W ∪ I);
/* Construction */
while (U 6= ∅)
∪ (W )};
pick v ∗ ∈ arg max{|N (v) ∩ U | : v ∈ U ∩ NG
if (D ∩ N (v ∗ ) 6= ∅)
pick u∗ ∈ D ∩ N (v ∗ ) and set parent(v ∗ ) ← u∗ ;
else
pick u∗ ∈ W ∩ N (v ∗ ) and set parent(v ∗ ) ← u∗ ;
end
I ← I ∪ {v ∗ };
D ← D ∪ {v ∗ } ∪ {u∗ };
W ← W ∪ (N (v ∗ ) ∩ U );
U ← U \ ({v ∗ } ∪ N (v ∗ ));
end
/* Improvement */
if (|D| = 2|I| − 1)
pick w∗ ∈ arg max{|N (w) ∩ I| : w ∈ W };
y ← the first vertex in N (w∗ ) ∩ I added to I;
D ← D ∪ {w∗ } \ ({parent(v) : v ∈ N (w∗ ) ∩ I} \ {parent(y)});
end
return D, I.
the first vertex for the independent set I and the connected dominating set D to be a
maximum degree vertex v ∗ in the graph. Then the remaining variables are initialized
correspondingly: W ← N (v ∗ ) and U ← V \ (W ∪ I).
Construction. We continue to construct our connected dominating set D and maximal
independent set I by recursively adding an unexplored vertex v ∗ from the neighborhood
of W that has the largest number of unexplored neighbors. In addition, to preserve the
connectivity of D we include one of the explored neighbors u∗ of v ∗ in D and assign
u∗ to be the parent of v ∗ . When selecting u∗ , the preference is given to a neighbor of
v ∗ which is already a parent of another vertex (if such a neighbor exists). This stage
terminates when there are no more unexplored vertices, at which point we will obtain
a maximal independent set I and a connected dominating set D.
Improvement. Finally, at the improvement stage we attempt to decrease the size of our
connected dominating set, in case a new parent vertex was used for each vertex that
was added to I at the construction stage, i.e. |D| = 2|I| − 1. This is done by adding to
D a vertex w∗ in W with the largest number of neighbors in I and removing from D
parents of all vertices from I that are adjacent to w∗ , except for the parent of the first
vertex from N (w∗ ) ∩ I that was added to I at the construction stage.
The proposed procedure is summarized in Algorithm 1.
Next, we analyze the performance of Algorithm 1. We will show that our algorithm
has a constant performance ratio.
6
Lemma 3 Algorithm 1 returns a connected dominating set D and a maximal independent set I.
Proof We first show that I is a maximal independent set. Indeed, at each step of our
algorithm v ∗ added to I is selected from set U , which does not have any neighbors
in I, therefore I is an independent set. Since the algorithm terminates when U = ∅,
every vertex from V will be either in I or the neighborhood of I, given by W , upon
termination, thus I is a maximal independent set. To show that the returned set D
is a connected dominating set, first observe that it contains the maximal independent
set I, which is also a dominating set. Therefore D is a dominating set. D is connected
throughout the execution of the construction stage, since every time we add v ∗ to D, it
is adjacent to its parent vertex u∗ , which is also added to D and is such that G[D∪{u∗ }]
is connected. Finally, we need to show that D is still a connected dominating set after
the improvement stage. Let w∗ be a vertex from W with the largest number of neighbors
in I, and let N (w∗ ) ∩ I ≡ {y1 , . . . , yq } (q = |N (w∗ ) ∩ I|) be the list of such neighbors,
specified in the order in which vertices y1 , . . . , yq were added to I at the construction
stage. Then y1 is exactly the vertex y picked at the improvement stage. Note that the
improvement stage is executed if and only each vertex in I has a different parent. Recall
that at the construction stage we select a parent for a vertex just added to I so that
the preference is given to an existing parent. Hence a different parent for each vertex in
{y1 , . . . , yq } is possible if and only if N (parent(y1 ))∩{y2 , . . . , yq } = ∅. This means that
parent(y1 ) has to be connected to some vertex from I \ {y1 , . . . , yq }. Therefore, if we
replace {parent(v) : v ∈ {y2 , . . . , yq }}) with w∗ in D, the set D will still be connected,
since vertices {y1 , . . . , yq } are all connected to w∗ and y1 is connected to the rest of D
through parent(y1 ). It also remains a dominating set, since I is still a subset of D.
Proposition 1 Algorithm 1 has the performance ratio of at most 22.
Proof We consider two cases, depending on whether the improvement stage of the
algorithm needed to be executed.
1. If after termination of the construction stage we have |D| < 2|I| − 1, then the
improvement stage is not executed and using Lemma 2 we obtain
|D| ≤ 2(|I| − 1) ≤ 2(11γc (G) + 1 − 1) = 22γc (G).
2. If after termination of the construction stage we have |D| = 2|I| − 1, we run the
improvement stage. There are two possibilities:
(a) max{|N (w) ∩ I| : w ∈ W } ≤ 2. Then using Lemma 2 with ν = 2 we obtain
|D| = 2|I| − 1 ≤ 2(2γc (G)) − 1 < 22γc (G).
(b) max{|N (w) ∩ I| : w ∈ W } ≥ 3. Then after performing the improvement stage,
we have
|D| = 2|I| − 1 − (|N (w∗ ) ∩ I| − 1) + 1 ≤ 2(|I| − 1)
and using Lemma 2 we obtain
|D| ≤ 2(|I| − 1) ≤ 2(11γc (G) + 1 − 1) = 22γc (G).
Thus, the statement is correct in all possible cases.
7
4 Complexity and approximation of the k-BCDS(∆) problem
We show that the approximation of the k-BCDS(∆) problem with a factor 2 − is
NP-hard. This also shows that the k-BCDS problem is NP-hard on general graphs.
Proposition 2 It is NP-hard to approximate the k-BCDS(∆) problem within a factor
2 − for any > 0.
Proof The reduction is from the minimum connected dominating set problem in general
graphs. Given a graph G = (V, E), we form a new edge-weighted complete graph
G0 = (V 0 , E 0 ) such that V 0 = V and the weight ce of an edge e ∈ E 0 is 1 if e ∈ E and 2
otherwise. Clearly, the edge weights of G0 satisfy the triangle inequality. Suppose that
we have a 2 − approximation algorithm, A, for the k-BCDS(∆) problem. We have the
following possibilities for the cost A(G0 ) of the solution obtained by applying A to G0 :
1. A(G0 ) = 2. Then the bottleneck cost of 2 is also the optimal solution. This leads
us to the conclusion that the minimum CDS in G is strictly greater than k.
2. A(G0 ) = 1. Thus there exists a CDS in G of size less than or equal to k.
Considering these two cases, the minimum connected dominating set problem can
be solved in polynomial time for any graph G (assuming that A is a polynomial-time
algorithm). Thus, it is NP-hard to approximate the k-BCDS(∆) problem within a
factor 2 − for any > 0.
Proposition 3 There exists a 3-approximation algorithm for the minimum k-BCDS(∆)
problem.
In order to prove this proposition, we present an approximation algorithm and show
that its performance ratio is at most 3. The method is presented in Algorithm 2 and
is based on the unified approach to approximating bottleneck problems developed by
Hochbaum and Shmoys [8]. We first sort the edges in nondecreasing order of their costs.
Next we solve the minimum bottleneck spanning tree problem, yielding the minimum
cost cē that guarantees the connectivity of the corresponding bottleneck subgraph. A
spanning tree T of G is a minimum bottleneck spanning tree if there is no spanning tree
T 0 of G with a cheaper bottleneck edge. It is well known that any minimum spanning
tree of a given graph G is also a minimum bottleneck spanning tree of G. Thus, we
can use a minimum spanning tree algorithm such as Prim’s algorithm or Kruskal’s
algorithm to find the minimum bottleneck spanning tree.
The cost cē of T is a lower bound on the optimal k-BCDS objective. We can
potentially improve this bound using the following argument.
Lemma 4 Let ce∗ be the optimal bottleneck cost for the k-BCDS(∆) on a graph G and
let the edge e = (u, v) have the largest cost, cmax , in G. Then we have:
ce∗ ≥ cmax /(k + 1).
Proof Since edge weights in G satisfy the triangle inequality, cmax cannot be greater
than the sum of weights of edges in any path between u and v in G(e∗ ). Let D be an
optimal k-BCDS in G. Then D is a connected dominating set in G(e∗ ), and there is a
path between u and v in G(e∗ ) that passes through vertices in D only. Since |D| ≤ k,
this path consists of at most k + 1 edges, each of cost at most ce∗ . Thus we have:
cmax ≤ (k + 1)ce∗ ⇒ ce∗ ≥ cmax /(k + 1).
8
Algorithm 2: The 3-approximation algorithm for k-BCDS(∆).
input a graph G = (V, E) with edge weight satisfying the triangle inequality, k;
sort all edges in E in nondecreasing order of their weights:
ce1 ≤ ce2 ≤ · · · ≤ cem , m = |V2 | ;
compute a minimum bottleneck spanning tree T of G with Cb (T ) = cē ;
let ei be the minimum cost edge such that cei ≥ max{cē , cmax /(k + 1)};
repeat
set Ĝ ← G(ei );
/* compute a maximal independent set I of Ĝ2 */
I ← {w}, where w is an arbitrary vertex in Ĝ;
while V \ N ∪2 [I] 6= ∅
Ĝ
choose u ∈ N ∪3 [I] \ N ∪2 [I];
Ĝ
Ĝ
S
I ← I {u};
end while
i = i + 1;
until |I| ≤ k;
return I.
Based on the above lemma, we form the bottleneck subgraph Ĝ = G(ei ) corresponding to the minimum cost edge ei in G such that cei ≥ max{cē , cmax /(k + 1)}. We
compute a maximal independent set I in Ĝ2 by starting from an arbitrary vertex and
recursively including a vertex u for which there exists v ∈ I such that dĜ (u, v) = 3 and
there is no v ∈ I such that dĜ (u, v) ≤ 2. We stop when it is not possible to expand
I. If |I| ≤ k, we can claim that we have a connected dominating set of size less than
or equal to k in the bottleneck subgraph G(e0 ), where ce0 ≤ 3cei . Otherwise, |I| > k
implies that G(ei ) does not have a connected dominating set of size at most k (as no
two vertices from k can be dominated by the same vertex), so we pick the next edge
weight in the sorted list and continue our search.
Proposition 4 For any given instance of the k-BCDS(∆) problem we have
Cb (I) ≤ 3Cb (D∗ ),
where D∗ denotes an optimal solution of k-BCDS(∆) problem for this instance.
Proof By the construction of set I and due to the triangle inequality, each time we add
a vertex u to I, there is a vertex v ∈ I such that for e = (u, v) we have ce ≤ 3cei . Thus,
if |I| ≤ k then I is a connected dominating set of size at most k in G(e0 ), where e0 is a
maximum cost edge in G with ce0 ≤ 3cei . It is easy to see that Cb (D∗ ) ≤ Cb (I), since
D∗ is an optimal solution. Furthermore, cei ≤ Cb (D∗ ), since for bottleneck costs less
than cei , the corresponding bottleneck subgraph cannot have a connected dominating
set of size k or less. Thus, we have:
Cb (I)/3 ≤ cei ≤ Cb (D∗ ) ≤ Cb (I)
5 Conclusion
We develop a constant-ratio approximation algorithm for the minimum connected dominating set problem in UBGs and introduce and study the bottleneck connected dom-
9
inating set problem in general and geometric graphs. The minimum k-BCDS problem
seeks a minimum edge weight in the graph such that the corresponding bottleneck
graph has a connected dominating set of size k. Motivated by the wireless network
applications, we propose this problem as a viable approach to determine an optimal
transmission range for a wireless network that guarantees a certain size of virtual
backbone modeled as a connected dominating set. We prove that it is NP-hard to approximate the k-BCDS with a factor of 2 − even for graphs whose edge weights satisfy
the triangle inequality. We propose a 3-approximation algorithm for graphs whose edge
weights satisfy the triangle inequality. A natural future research direction is to design
more practical, distributed, algorithms for solving the k-BCDS problem in unit ball
graphs.
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