In this paper, we are interested in a variant of the dominating set problem: the edge monitoring problem. The edge moniroring (or watchdog technique) is a mechanism for the security of wireless sensor networks BMVH; GZYN; DXYCX. The basic idea is to select some nodes as monitors in a given sensor network. These monitors are employed for carrying out monitoring operations by promiscuously listening to the transmission of two nodes. They can also perform basic operations of communication and sensing in the network.
The edge monitoring problem is defined as follows. Let be a graph and be an integer weight function on edges of . An edge monitoring set of is a set of vertices such that each edge of is monitored by at least vertices of the set . A node monitors an edge if its both end-nodes are neighbors of i.e., together with form a triangle in the graph. Consider the example in Figure 1. The black nodes can monitor all edges depicted in bold.
Dong et al. DXYCX proved that the edge monitoring problem is NP-complete even restricted to unit disk graphs and they propose a polynomial-time approximation scheme for this class of graphs. Baste et al. parametrizedplanar2016 focused on parametrized complexity. They proved that the problem is -hard on general graphs and proposed an FPT algorithm for planar graphs and, more generally, for apex-minor-free graphs.
This paper focuses on the complexity of the edge monitoring problem and its weighted version on different classes of graphs. A weighted version of the edge monitoring problem is applied on graphs with weights on vertices (in addition to weights on edges). Let be a weighted graph with the weight associated to a vertex . The aim is to find a set that monitors and minimizes .
Among the classes studied in this paper, we consider block graphs, split graphs, cographs and interval graphs which are perfect graphs. Note that the class of complete graphs is included in all graph classes mentioned before. Since we prove that the edge monitoring problem is hard for complete graphs, we consider the problem in these classes with more restricted conditions. We also have a special interest in the unit disc graphs and planar graphs.
This paper is organized as follows. Section 2 gives formal definitions of the problem and its variant. Some basic graph terminologies and concept of complexity are also presented. Section 3 presents some introductory results. In Section 4, we study the problem in complete graphs and block graphs. We give a polynomial time approximation scheme for weighted complete graphs. Sections 5,6,7 are dedicated to interval graphs, cographs and split graphs respectively. In section 8, we prove that the problem is NP-complete on planar unit disk graphs. Besides, we show that there exists a PTAS for Weighted Edge Monitoring on weighted planar graphs and more generally on weighted apex-minor-free families of graphs. The last section summarizes all results of this paper and give some suggestions for further research.
In this section, we give some basic graph terminology and complexity used throughout this paper. We also give definitions of the edge monitoring problem and all concepts used around this problem.
2.1 Basic notions of graphs
Graphs considered in this paper are simple, undirected and without loops. Let be a graph. The (open) neighborhood of a vertex is . The closed neighborhood of is . For a set , . The induced graph of by , denoted by contains all the edges of whose extremities belong to . A clique is a set such that each two vertices of are adjacent. An independent set is a set such that no edge of has its two end vertices in . The clique number of , denoted by , is the cardinality of a maximum clique in . A graph is chordal if it has no induced cycle of length more than . The treewidth of , denoted by , is . A set is a dominating set of if . A set is a total dominating set of if . a set is a double dominating set of if for every vertex , . ) (resp. , ) denotes the size of a smallest dominating set (resp. total dominating set, double dominating set) of or if such a set does not exist.
2.2 Edge monitoring
Let be an edge of a graph . We denote by the set of vertices such that forms a triangle. We say that monitors . Let be an integer. A set -monitors an edge if . Let be a graph and be a weight function over the edges of . monitors if -monitors every edge in . The couple is called a weighted graph. (and if no monitoring set exists). where is 1-uniform.
We define the problem EdgeMonitoring as a decision problem. However, we use the same name for the minimization problem and the parameterized version with as parameter.
Let such that is a graph, and . . is also called a weighted graph. Similarly to EdgeMonitoring, we define the problem WEM.
Let be a weighted graph with . Then . Whenever and are obvious from the context, we write instead of . A family of weighted graphs is -bounded if there exists an integer such that for every .
Let be a minimization problem. Let . An algorithm is called a -approximation algorithm for , if, for all instances of , it delivers a feasible solution with objective value such that . A polynomial time approximation scheme (PTAS for short) for is a family of -approximation algorithms computable in polynomial time in the input size for any .
Parameterized complexity consists in studying the complexity of problems according to their input size, but also to another parameter. For any basic notions of parameterized complexity (, FPT-reduction, etc.); see Flum:2006.
In the folowing, we prove that 1-uniform EdgeMonitoring cannot be approximated with a constant ratio. We use a reduction from this problem.
1-uniform EdgeMonitoring cannot be approximated within for any , unless .
It has been proved in chlebik2008approximation that TotalDominatingSet cannot be approximated within for any , unless . We will define an approximation preserving reduction from TotalDominatingSet to 1-uniform EdgeMonitoring. Let be a graph without isolated vertex. We construct from by adding three vertices which form a clique and connecting to every vertex in . We will prove that .
Let be a total dominating set of and . Then is a monitoring set of . Indeed, the edges , and are monitored by , and respectively. The edges in are monitored by . Let be a vertex in then has a neighbor in . Thus, is monitored by .
Now, let be a monitoring set of . . Otherwise, , or is not monitored by . Let . We will prove that is a total dominating set of . Let be a vertex of . The edge is monitored by a vertex in . Since forms a triangle, is adjacent to a vertex in . Hence, .
Using the same method as in Theorem 1 of klasing2004hardness we obtain the desired result.
3 Complete graphs and block graphs
In this section we present some results of WEM problem on complete graphs and block graphs.
A block graph is a graph where each biconnected component (block) is a clique. The block-cut tree of a connected graph is defined as follows. The vertices of are the blocks and the articulation points of . There is an edge between an articulation point and a block in if .
Let be a weighted graph such that is a complete graph, and . Then, . Moreover, every set with is a monitoring set of .
Since there exists an edge of weight , we need vertices to monitor it. Thus, . Let be a set such that . Then, every edge is -monitored by . Indeed, let . Then, the set of size at least -monitors .
Let be a weighted graph such that is a complete graph and is k-uniform with and . Then, .
Assume, for the sake of contradiction, that there exists a set that monitors such that . If , let be the unique element of . Let an edge incident to . Then, is not -monitored by . Otherwise, let and be two elements in . Then, so is not monitored by .
EdgeMonitoring is NP-complete on complete graphs. Moreover, EdgeMonitoring is -complete on complete graphs.
We will prove that EdgeMonitoring is equivalent to IndependentSet under FPT-reductions. Since IndependentSet in -complete, the results follow.
First, we show a reduction from IndependentSet to EdgeMonitoring. Let be an instance of IndependentSet. Without loss of generality, we can assume that is connected. Indeed, it is easily seen that IndependentSet remains -hard under this restriction. We build an instance of EdgeMonitoring as follows: is a complete graph and for each edge , we have if and otherwise.
We show that is a positive instance of IndependentSet if and only if is a positive instance of EdgeMonitoring. First of all, notice that there is no monitoring set of size less than . Indeed, assume, for the sake of contradiction, that there is a monitoring set of size less than . Since is connected, there exists an edge incident to a vertex in and such that . We have so there is a contradiction.
Now, let such that . Then, we have:
is a monitoring set of iff for each , iff for each in E, iff is a stable of .
Now, we show a Turing FPT-reduction from EdgeMonitoring to IndependentSet. The reduction is presented in Algorithm 1. Notice that this algorithm is recursive.
First, let us prove that admits a monitoring set of size at most if Algorithm 1 returns True. We proceed by induction on . If , it is clear that Algorithm 1 returns True if and only if . Now, assume that . If Line 6 returns True then admits a monitoring set of size at most by induction hypothesis. Assume now that Line 11 returns True. Then, there exists an independent set of size in . Thus, is a monitoring set of . Indeed does not admit an edge with by Lines 2-3. Edges with have no extremities in by construction of . Hence, these edges are monitored by . Edges with have at most one extremity in also by construction of . Thus, these edges are monitored by . Edges with are necessarily monitored by since .
Now, let us prove that Algorithm 1 returns True if admits a monitoring set of size at most . We proceed by induction on . If then necessarily . Thus, Algorithm 1 returns True. Now, assume that . If then Algorithm 1 returns True in Line 6 by induction hypothesis. Assume now that then it is easily seen that is an independent set of with . Then Algorithm 1 returns True in Line 11. This completes the proof.
WEM can be solved in polynomial time on -bounded weighted complete graphs.
Let with a complete graph. By Lemma 2, . Therefore, it suffices to enumerate all sets that monitor and such that . There are such sets. Thus, the problem can be computed in polynomial time.
WEM can be solved in quasi-linear time on uniform complete graphs.
The following lemma is useful to establish the connection between of a graph and of its -connected components.
Let be a weighted graph, and two graphs and such that , and . Let . Let obtained from by replacing the weight of by . Then .
Let be optimal solutions of , , respectively.
We first prove : if then is a solution of having weight . If then is a solution of having weight . Thus we have .
Now we prove : let be an optimal solution of . We have and are solutions of and respectively. We have to consider two cases:
: We have and by optimality of and . Since , .
: This implies that . Since and , then
Consequently we have . This completes the proof of the lemma.
The two statements hold:
WEM can be solved in polynomial time on -bounded weighted block graphs.
WEM can be solved in quasi-linear time for block graphs where is uniform.
Without loss of generality, we can assume that is connected. We will prove the first statement. The proof of the second statement is similar. Let be a -bounded weighted block graph. We first compute the block-cut tree of . This can be done in linear time hopcroft1973algorithm. Then, we choose a clique that corresponds to a leaf of and the articulation point that is neighbor of in . Let and . is also a block graph. Thus, we can apply Lemma 7. It suffices to compute , and . can be computed in polynomial time by using Lemma 5. Proof of Lemma 5 can be easily modified to compute . can be computed by induction.
4 PTAS for the Wem problem in weighted complete graphs
In this section, we study the approximation complexity of the weighted monitoring set
problem in vertex-weighted complete graphs.
There exists a PTAS for WEM on complete graphs.
Fix and . Let such that is a complete graph and . Let OPT denote an optimal solution for .
We have to consider three different cases:
Case 1. :
Using Lemma 2, we have . We just need to enumerate all the sets with size at most . We can do it in polynomial time .
Case 2. :
Clearly, there exists no monitoring set for since there exists an edge such that and .
Case 3. and :
Let be the set of the first vertices sorted in ascending order by weight . Let be the set of sets such that and . We prove that has a polynomial size. Indeed, we have
Since for every , it holds . Thus is polynomial in .
The algorithm consists to enumerate all the sets in and take a solution of minimum weight. This can be done in polynomial time. We distinguish two subcases as follows:
Case 3.a. :
Clearly, the algorithm returns an optimal solution.
Case 3.b. :
Notice that is a (non necessary optimal) solution by Lemma 2 and the algorithm returns a solution such that . We will prove that . Let denote the vertices in . Let denote the vertices in . Let denote the vertices in sorted in ascending order by weight . Since , we have . In the following, we will bound the approximation ratio of the solution:
(3) (4) (5) (6) (7) (8)
5 Interval graphs
In this section, we give a polynomial algorithm for computing WEM on weighted interval graphs. This algorithm uses dynamic programming. First, we introduce some definitions.
A graph is an interval graph if there exists intervals of the real line such that if and only if for every distinct vertices . We say that is a realization of . Without loss of generality, we can assume that there are no intervals and that have a common extremity.
Given an interval graph and a realization , we define a total order (resp. ) over such that (resp. ) if (resp. ).
The following definition is a refinement of the nice tree decomposition introduced by Kloks kloks94
nicepath-decomposition Let be an interval graph and be a realization of . A nice path decomposition of is a sequence of sets of vertices such that
all sets are cliques of ;
every edge appears in a set ,
for every vertex , the set of indices such that is a segment of .
For every ,
( introduces the vertex )
or ( forgets the vertex ).
the order in which vertices are introduced corresponds to
the order in which vertices are forgotten corresponds to
nicepath-decomposition Let be an interval graph and be a realization of . Then has a nice path-decomposition that can be computed in linear time.
For the next lemmas, we consider an interval graph and a nice path-decomposition of . Moreover, we introduce the following notations. For , is the set of vertices appearing in some set , , but not in . and .
A set is an -partial solution if every edge in that has an extremity in is -monitored by . The -representant of , denoted by , contains exactly the greatest vertices in w.r.t. or is if . We say that extends if is the -representant of .
We denote by the set of -representants of -partial solutions. is a function such that .
Before presenting the algorithm, we introduce two lemmas. The second is the key of the algorithm.
Let , such that and . Then .
Let , and the intervals that represent , and respectively in the realization of . Since and , and . Since , we have and since , we have . Thus . Consequently, .
Let , , and . If is -monitored by then is -monitored by .
First, notice that . If , then and the lemma is trivially verified. Now, assume that and let . By Lemma 11, every vertex belongs to and . So all elements in except at most two ( and ) belong to . Thus and is -monitored by .
To solve WEM on interval graphs, a naive algorithm consists to iterate over the sets and to compute for each the set of -partial solutions. Unfortunately, the algorithm is non polynomial since the set of -partial solutions can be exponential. The key of the algorithm is as follows: instead of considering all the -partial solutions, we consider the representants of the -partial solutions. Since the number of representants is polynomially bounded by , the algorithm will run in polynomial time. Lemma 12 guarantees that we don’t miss solutions. Indeed, let be an -partial solution. If introduces the node , then and are -partial solutions. If forgets the node then is an -partial solution if and only if every forgotten edge (i.e. an edge having as extremity and the other extremity in ) is -monitored by . But thanks to Lemma 12, it suffices to check that these edges are -monitored by .
We present now Algorithm 2.
The next lemma shows that the sets and functions computed by Algorithm 2 correspond to the sets and functions defined previously.
For every , after the run of Algorithm 2, we have and for every .
We prove by induction on . The property is clearly verified for . Now, suppose that the property holds for and prove it for .
and for each , : let . We consider two cases.
forgets the vertex : then, comes from some such that , and is added to by Lines 9-11. Using the induction hypothesis, and . Let be a -partial solution of weight that extends . By Line 8 of the algorithm, every edge where is -monitored by and thus by . Consequently, is an -partial solution with . Thus, and .
introduces the vertex : There are two possibilities.
: then comes from some such that , and is added to by Lines 14-16. By induction hypothesis, and . Let be a -partial solution of weight that extends . is an -partial solution with