Memetic Algorithm with Filtering Scheme for the Minimum Weighted Edge Dominating Set Problem

The minimum weighted edge dominating set problem (MWEDS) generalizes both the weighted vertex cover problem and the problem of covering the edges of graph by a minimum cost set of both vertices and edges. In this paper, we propose a meta heuristic approach based on genetic algorithm and local search to solve the MWEDS problem. Therefore, the proposed method is considered as a memetic search algorithm which is called Memetic Algorithm with filtering scheme for minimum weighted edge dominating set, and called shortly (MAFS). In the MAFS method, three new fitness functions are invoked to effectively measure the solution qualities. The search process in the proposed method uses intensification scheme, called “filtering”, beside the main genetic search operations in order to achieve faster performance. The experimental results proves that the proposed method is promising in solving the MWEDS problem. (Less)


I. INTRODUCTION
The Minimum Edge Dominating Set (MEDS) is a subset of edges of minimum cardinality, where each edge is be in the edge dominating set, or adjacent to some edges in the edge dominating set [11], [12], [24].The weighted version of MEDS seeks to find an edge dominating set with a minimum total weight [9].The MEDS problem is a hard combinatorial problem, classified as NP-hard [24], and in general cannot be solved exactly in polynomial time.The MEDS problem is one of the fundamental covering problems in graphs; edge cover, vertex cover, dominating set and edge dominating set.These domination problems in graphs have been subject of many studies in graph theory, and have many applications in operations research, resource allocation and network routing, as well as in coding theory [4], [11], [12], [25].
There are many algorithms proposed for solving MWEDS.Although these algorithms guarantee the optimality of the solutions they find, they may fail to give a solution within reasonable time for large instances.As the size of the problem increases, these methods become futile.Meta-heuristics are powerful search methods which can be efficiently in providing satisfactory solutions to large and complex problems such as vertex cover [20], dominating set [14] and edge coloring [16] in a reasonable time.However, up to the authors' knowledge, there are no studies up to day used meta-heuristic techniques for solving the MWEDS problem.
Genetic Algorithms (GAs) are the most popular metaheuristic algorithms that have been employed in wide variety of problems [3].Actually, GAs are able to incorporate other techniques within its framework to produce a hybrid method that brings more promising one.One direction of such hybridization is to use local search which can accelerate the search process in a pure GA.This modification yields another search approach which is called the Memetic Algorithm (MA) [17].
Several meta-heuristic methods have been developed to solve different problems in graph theory and combinatorial optimization [2], [19].However, the number of contributions that deal with the graph domination problems is very limited.In this paper, we propose a memetic algorithm with filtering scheme for finding the minimum edge dominating set, called shortly MAFS.It uses a 0-1 variable representation of solutions in searching for the MWEDS, and invokes three new fitness functions to measure the solution qualities.Intensification search and filtering schemes are used beside local search in order to enhance the performance of the MAMEDS method.
The paper is organized as follows.The next section gives a brief description about the MWEDS problem as preliminaries needed throughout the paper, and highlights the related works in solving the considered problem.Section 3 describes the proposed method steps in details.Section 4 reports numerical experiments and results.Finally, the conclusions make up Section 5.

II. PROBLEM FORMULATION AND RELATED WORKS
Given an undirected weighted graph G = (V, E, W ), without loops and multiple edges, where V is the set of nodes (or vertices), E the set of edges, and W is the set of positive edge weights represented by variables w 1 , w 2 , . . ., w m , (where each w i corresponds to an edge i = (u, v) ∈ E).An edge (u, v) of G is said to dominate itself and any edge adjacent to it in G.An edge dominating set (EDS) is a set of edges which is collectively dominate all the other edges in the graph G.The Minimum Weight Edge Dominating Set (MWEDS) problem seeks to find an edge dominating set EDS of minimum total weight Σ e∈D w(e).
The edge dominating set problem is a basic problem introduced in Garey and Johnson's work [10] on NP-completeness.Yannakakis and Gavril [24] proved that the edge dominating set problem is NP-hard even in planar or bipartite graphs www.ijarai.thesai.org of maximum degree 3.Although the EDS has important application in areas such as telephone switching networks, a very little work was known about the weighted version of the problem.
For the EDS problem, Randerath and Schiermeyer [8] presented the first exact algorithm of time complexity O(1.4423 n ) algorithm and Fomin et al. [6] improved this to O(1.4082 n ).Rooij and Bodlaender [21] got an O(1.3226 n ) algorithm by using the "measure and conquer" method, which was further improved to O(1.3160 n ) [23], where n is the number of vertices.From the point of approximation algorithm, the best known result was proposed in [18], which gave a 2-approximation algorithm for WEDS problem.Recently, parameterized computation theory was applied to solve the EDS and WEDS problems.Fernau [5] presented parameterized algorithms of time complexity O(2.62 k ) for EDS and WEDS problem respectively.The above result was further reduced by Fomin [7], which gave a parameterized algorithm of time O(2.4181 k ).For the first time Wang [22] presented an enumeration algorithm of time complexity O(5.6 2k k 4 z 2 +4 2k nk 3 z) for WEDS problem.Although these algorithms provide the optimal solution, they are too slow on graphs with few hundreds of nodes.Therefore, when deals with a very large graphs, these algorithm become impractical.This motivates us to consider meat-heuristics to design more efficient algorithm to solve the MWEDS problem.

III. PROPOSED METHOD
In this section, we describe the components of the MAFS method, and then state its formal algorithm at the end of this section.The MAFS method is an evolutionary algorithm, therefore, we first start by describing the solution representation and the fitness function.Then, the genetic operators; selection, crossover and mutation are defined.The main memetic search element, local search, is stated after that.Finally, our intensification schemes are explained.

A. Graph Representation
The graph represented as n V × n V adjacency matrix A, where n V is the number of vertices in the graph.The nondiagonal entry a ij = w e , where w e is an integer weight associated with each edge e connected the vertex i to vertex j.Form an adjacency matrix we create edges matrix E m which include all edges in the graph.Edge matrix dimension is n E × 3, where n E is the number of edges in the graph.The first two columns are the vertex numbers which represent the endpoints of edges and the third columns represent the weights of each edge in the graph.

B. Solution Representation
A solution s will be represented as a bit vector with size equal to the number of edges in the graph.Therefore, s is equal to (s 1 , s 2 , . . ., s nE ), as shown in Figure 1.The subscript numbers 1, 2, . . ., n E , are related to the corresponding edges in E m .If a component s i of s, i = 1, . . ., n E , has the value 1, then the edge represented by the i-th row in E m is contained in the edge subset represented by solution s.Otherwise, the solution s does not contain the i-th edge.

C. Fitness Function
Fitness function fit is a function designed to measures the quality of a solution which plays a major role in the selection process.The main idea in designing the fitness function is that better solutions will have a higher fitness function value than worse one.Three fitness functions are invoked to effectively measure the solution qualities.
where 0 ≤ α ≤ 1, κ > 1 is an integer, and ρ d , T sum (s) and sum w (s) are calculated by where n D is the number of edges dominated by the subset of edges D represented by the solution s and n E the number of edges in the graph.All three fitness function consist of two parts, the first part n D /n E , reflects the size of domination on G by s.If s represents an edge dominating set, then this part is equal to 1. On the other hand, the second part distinguishes between solutions that have the same values of the first part based on the sum of weights associated with each edge contained in each of them.It is worthwhile to mention that the second term is designed to make f it(s 1 ) < f it(s 2 ) in only two cases: , where x 1 and x 2 are the numbers of edges covered by s 1 and s 2 respectively, or The parameter κ is set equal to 4 to highly distinguish between solutions that have the same domination number.www.ijarai.thesai.org

D. Genetic Operators
The parent selection mechanism first produces an intermediate population, say P ′ from the initial population P : P ′ ⊆ P as in the canonical GA.For each generation, P ′ has the same size as P but an individual can be present in P ′ more than once.The individuals in P are ranked with their fitness function values based on the linear ranking selection mechanism [1], [13].Indeed, individuals in P ′ are copies of individuals in P depending on their fitness ranking: the higher fitness an individual has, the more the probability that it will be copied is.This process is repeated until P ′ is full while an already chosen individual is not removed from P .
The crossover operation has an exploration tendency, and therefore it is not applied to all parents.First, for each individual in the intermediate population P ′ , the crossover operation chooses a random number from the interval (0, 1).If the chosen number is less than the crossover probability p c ∈ (0, 1), the individual is added to the parent pool.After that, two parents from the parent pool are randomly selected and mated to produce two children c 1 and c 2 , which are then replacing their parents in P ′ .These procedures are repeated until all selected parents are mated.The standard one-point crossover [15] is used in MAFS to compute children.
For each gene each in all individuals in the intermediate population P ′ , a random number from the interval (0, 1) is associated.If the associated number is less than the mutation probability p m , then the individual is mutated using the standard uniform mutation operation [15].

E. Local Search
In LocalSearch mechanism, we add or delete some edges to improve the best solution s best found so far, and this process is repeated n l times.The formal description of this mechanism is shown in Procedure 1.

Procedure 1: (LocalSearch)
1) Set a suitable value to n l .2) Repeat the following Steps (2)(3)(4)(5)(6)  In our numerical experiments, the number n l of local search iterations is set equal to 0.1 × n E in order to save computational costs.

F. Intensification Schemes
The intensification mechanism which called "Filtering" is used in MAFS to reduce the cost of the solution computed.This mechanism basically checks if an edge contained in s best can be removed without losing the coverage.

G. MAFS Algorithm
MAFS starts with an initial population of chromosomes P 0 generated randomly.Each chromosome represents a trial solution to the MWEDS problem.During each generation, the quality of each chromosome in the population is evaluated by using three fitness functions (see Equations 1,2 and 3).MAFS applies Procedure 1 to improve the best solution.
In each generation, the population is updated through genetic operators.Linear ranking selection algorithm uses to select parents for standard one-point crossover and uniform mutation to generate members of the new population [14] Otherwise, set t := t + 1, and go to Step 3.

IV. NUMERICAL EXPERIMENTS
The MAFS algorithm was programmed using MATLAB.In this experimental section, we technically discuss the www.ijarai.thesai.orgimplementation of the MAFS code as well as its results.This section also shows how the test graphs used in the numerical simulations are generated.

A. Graph Generation
In order to measure the performance of MAFS we apply it on number of graphs with different sizes.The previous works in solving MWEDS did not implemented for special types of graphs.Thus, the graphs which we used in our experiments are randomly generated with a known edge domination number γ(G) and optimal total weight op w .The following algorithm describe how these graphs are constructed.
, and the number of edges n E = max E × d, where d is the density of edges in the graph which is set to be in (0, 1), and n V is the number of vertices.2) Divide the vertices into two groups: -V ED with size equal to γ(G) × 2, and has vertices incident to dominant edges.Therefore, each pair of them is connected.
and has vertices not incident to dominant edges.3) Add edges to connect the graph vertices to reach the edge density d.This edge adding process should satisfy the following condition in order to maintain the edge domination number equal to γ(G).
-No edge connects two vertices belong to different pairs in V ED .4) Set the weights w e randomly for each edge in G such that 0 < w e ≤ l 1 for each dominant edge and l 1 < w e ≤ l 2 for the remaining edges.
In our numerical experiments, the parameters l 1 and l 2 is set equal to γ(G) and n E , respectively.
MAFS was applied to 15 instances of MWEDS problems created from the five graphs G1-G5, see Table I.These three graphs generated randomly with a number n V of vertices and different number n E of edges depending on the density number d for each instance.For each problem instance, the edge domination number γ(G) and the optimal total weight op w was known and the code was run 10 times.

B. Parameter Setting
Table II summarizes all parameters setting used in MAFS with their assigned values.These chosen values are based on our numerical experiments.

C. Comparison Results
In this section, we study the performance comparison of the proposed MAFS with three fitness functions that we introduce in Equations 1, 2 and 3. We have two comparison results, the first comparison of MAFS with (f it 1 ) against MAFS with (f it 2 ), and the results of this comparison are reported in Table  ).This measure gives the average of the optimal solution values found in the independent runs.

2) Rate Number (rate). The rate shows how many times
MAFS acquires an optimal solution op w .

1) Performance Comparison of MAFS with (f it 1
) and (f it 2 ): The results of this comparison are reported in Table III.The results show that MAFS with (f it 1 ) could not acquire the optimal total weight op w for all instances of the MWEDS problem especially when the number of edges increased proportionally with the graph size.MAFS with (f it 2 ) achieve significant improvement in the average results and in acquiring the optimal total weight op w for all instances.However it has a low rate (rate) for instances with large number of edges.
2) Performance Comparison of MAFS with (f it 2 ) and (f it 3 ): In this comparison, we compared MAFS with (f it 2 ) against MAFS with (f it 3 ).The results of this comparison are reported in Table IV.To achieve the best performance of the MAFS, the f it 2 was moderated by adding weights α, and (1 − α) to get a new fitness function f it 3 in 3. The weight parameters α is set equal to 0.4, which is used to efficiently trade-off between the trail solutions.The comparison results confirm a superior performance of MAFS with f it 3 in both terms (Ave.) and (rate) against the other two fitness functions.
In Table V the instances generated by algorithm 4 with modifications in the role of edge weights such that the dominant edges assigned weights w e from {1, 2, ..., γ(G)}, and for the remaining edges the set {γ(G) + 1, γ(G) + 2, . . ., γ(G) + 30}.MAFS with f it3 applied for these instances.The results show that when the dominant edges have different weights scheme used beside the genetic and local search methodologies in order to achieve better performance.Three new fitness functions invoked to maximize the performance of the proposed method.These fitness functions consider different ways to balance between two objectives; edge dominating and weight minimizing.Specifically, two of these fitness functions use absolute additions of valued functions that measure the considered objectives while the third one uses a weighted addition way.Numerical experiments of MAFS using the three fitness functions on various test graphs show that the MAFS with a weighted fitness function outperform the MAFS with the other two fitness functions.In addition, the proposed method show very promising performance to obtain minimum weighted edge dominating sets for different graphs used in the numerical experiments.
x nE } of all positions of value one in s best .3) Repeat the following Steps (4-5) for j = 1, ..., n E .4) Set s best xj = 0, and compute the new fitness value.5) If the fitness value is increased, update s best .
. MAFS invokes Local Search Procedure to update the current population.If a certain number of consecutive generations without improvement is achieved, MAFS invokes Procedure 2 to improve the best edge dominating set s best obtained so far, if it exists.The search will be terminated if the number of generations exceeds g max , or the number of consecutive generations without improvement exceeds a pre-specified number.Initialization.Set values of P size , g max .Set the crossover and mutation probabilities p Local Search.Evaluate the fitness function of all chromosomes in P 0 by using the Equations 1, 2 or 3, and then apply Procedures 1 to improve the best trial solution in P 0 .Set the generation counter t := 0. 3) Parent Selection.Select an intermediate population Ṕt from the current population P t using the linear ranking selection.4) Crossover.Apply the standard one-point crossover to chromosomes in Ṕt , and update Ṕt .5) Mutation.Apply the standard uniform mutation to chromosomes in Ṕt , and update Ṕt .6) Survival Selection.Evaluate the fitness function of all generated children in the updated Ṕt , and set P t +1 = Ṕt .If the best solution in P t +1 is worse than the best solution in Ṕt , then replace the worst solution in Ṕt +1 by the best solution in Ṕt .7) Local Search.Apply Procedure 1 to improve the s best , update WEDS.8) Filtering.If s best represents a weight edge dominating set, then apply Procedure 2 to improve it, update WEDS.9) Stopping Condition.If t > g max , then terminate.
c ∈ (0, 1) and p m ∈ (0, 1), respectively.Set WEDS to be an empty set.Generate an initial population P 0 of size P size .2)

TABLE II
The second performance comparison of MAFS with (f it 2 ) against MAFS with (f it 3 ), and the results of this comparison are reported in TableIV.To measure the performance of each method, two quantities are used in the comparisons which are computed as follows.