Minimum Weight Dominating Set Problem Research Articles

Chemical Reaction Optimization for Minimum Weight Dominating Set

Dominating set of a graph can be defined as the set of vertices that can cover all other vertices of the graph. The minimum weight dominating set (MWDS) is the minimum number of vertices in the dominating set with minimum total weight. In recent times, the chemical reaction optimization algorithm (CRO) has shown its supremacy in solving these types of problems. Therefore in this paper, a novel approach based on CRO has been proposed to solve the MWDS problem. The proposed method uses a repair-based technique to generate a molecule. To make the solution feasible by covering all vertices and to get better results, three supporting operators are implemented along with the CRO operators. Besides this, two repair operators are introduced. In the first repair operator, the searching procedure works based on the scaling properties of vertices, and the second one is a unique method for eliminating common neighbors of vertices of the dominating set. The performance of the proposed method is better than any other existing related algorithms. The performance is measured from different graphs of the benchmark datasets. It can be mentioned that the proposed method takes minimal running time to obtain the minimum weight compared to other benchmark methods.

Improved local search for the minimum weight dominating set problem in massive graphs by using a deep optimization mechanism

The minimum weight dominating set (MWDS) problem is an important generalization of the minimum dominating set problem with various applications. In this work, we develop an efficient local search scheme that can dynamically adjust the number of added and removed vertices according to the information of the candidate solution. Based on this scheme, we further develop three novel ideas to improve performance, resulting in our so-called DeepOpt-MWDS algorithm. First, we use a new construction method with five reduction rules to significantly reduce massive graphs and construct an initial solution efficiently. Second, an improved configuration checking strategy called CC2V3+ is designed to reduce the cycling phenomenon in local search. Third, a general perturbation framework called deep optimization mechanism (DeepOpt) is proposed to help the algorithm avoid local optima and to converge to a new solution quickly.Extensive experiments based on eight popular benchmarks of different scales are carried out to evaluate the proposed algorithm. Compared to seven state-of-the-art heuristic algorithms, DeepOpt-MWDS performs better on random and classic benchmarks and obtains the best solutions on almost all massive graphs. We investigate three main algorithmic ingredients to understand their impacts on the performance of the proposed algorithm. Moreover, we adapt the proposed general framework DeepOpt to another NP-hard problem to verify its generality and achieve good performance.

HSLS: An efficient local search algorithm for the hardware and software partitioning problem

Hardware and software partitioning (HW/SW) is a crucial step in the co-design of embedded systems, which is a typical combinatorial optimisation problem. In practice, HW/SW instances would be of large scale, so it is of significant importance to design effective heuristic algorithms to obtain a high-quality partitioning scheme. Different from previous heuristics, the HW/SW problem is treated as the minimum weight dominating set (MWDS) problem in this letter, and the mature local search framework for solving the MWDS problem is introduced to solve the HW/SW problem. In addition, two suitable vertex selection rules based on scoring function and tabu are proposed. Combined with the introduced local search framework, an efficient local search algorithm named HSLS is obtained. 96 classic benchmark instances are used to evaluate the performance of the HSLS algorithm, and the four latest evolutionary algorithms designed for the HW/SW problem are used as comparison algorithms, which are genetic algorithm, binary particle swarm optimisation, differential evolution, and group theory-based optimisation algorithm. The experimental results show that the HSLS algorithm is far superior to the four comparison algorithms in terms of the optimal solution and obtains optimal solution on 78 benchmark instances, the four comparison algorithms obtain 43, 29, 42, and 41 optimal solutions respectively.

Swarm intelligence, exact and matheuristic approaches for minimum weight directed dominating set problem

There exist numerous heuristic and exact approaches in the literature for addressing minimum weight dominating set problem (MWDS) on vertex-weighted undirected graphs which is a well known NP-hard problem. However, little attention has been paid to its counterpart in vertex-weighted directed graphs called minimum weight directed dominating set problem (MWDDS) despite its use in modeling real-world applications involving directed interactions. As directed graphs can model undirected graphs, MWDDS can be considered as a generalization of MWDS, and hence, MWDDS is also NP-hard. In this paper, we present two approaches based on swarm intelligence, one approach based on integer linear programming (ILP) and one matheuristic approach to address the MWDDS. These approaches are the first approaches for MWDDS in their respective categories. One of our swarm intelligence approach is based on artificial be colony (ABC) algorithm, whereas the other is based on invasive weed optimization (IWO) algorithm. Both these approaches are hybridized with problem specific heuristics and a local search mechanism. We have evaluated the performance of our approaches on benchmark instances derived from the standard benchmark instances of MWDS. Computational results show the effectiveness of our approaches.

An evolutionary algorithm based hyper-heuristic framework for the set packing problem

In recent years, hyper-heuristics have received massive attention from the research community as an alternative of meta-heuristics. In a hyper-heuristic, generation or selection of an effective heuristic among a pool of heuristics is an important and challenging task in the search process. At each iteration, a suitable heuristic can take the search process toward the global optimal solution. Moreover, some additional factors such as quality and the number of heuristics also affect the performance. In this paper, we propose an evolutionary algorithm based hyper-heuristic framework that incorporates dynamic selection of parameters. To test its generality, effectiveness and robustness, we apply this approach on two different NP-hard problems - set packing problem (SPP) and minimum weight dominating set (MWDS) problem. The proposed approach for the SPP and the MWDS problem has been evaluated respectively on their respective set of benchmark instances. Computational results show that the proposed approach for the SPP and MWDS problem perform much better than their respective state-of-the-art approaches in terms of the solution quality and computational time.

An efficient matheuristic for the minimum-weight dominating set problem

A minimum dominating set in a graph is a minimum set of vertices such that every vertex of the graph either belongs to it, or is adjacent to one vertex of this set. This mathematical object is of high relevance in a number of applications related to social networks analysis, design of wireless networks, coding theory, and data mining, among many others. When vertex weights are given, minimizing the total weight of the dominating set gives rise to a problem variant known as the minimum weight dominating set problem. To solve this problem, we introduce a hybrid matheuristic combining a tabu search with an integer programming solver. The latter is used to solve subproblems in which only a fraction of the decision variables, selected relatively to the search history, are left free while the others are fixed. Moreover, we introduce an adaptive penalty to promote the exploration of intermediate infeasible solutions during the search, enhance the algorithm with perturbations and node elimination procedures, and exploit richer neighborhood classes. Extensive experimental analyses on a variety of instance classes demonstrate the good performance of the algorithm, and the contribution of each component in the success of the search is analyzed.

A Binary Particle Swarm Optimization for the Minimum Weight Dominating Set Problem

The minimum weight dominating set problem (MWDSP) is an NP-hard problem with a lot of real-world applications. Several heuristic algorithms have been presented to produce good quality solutions. However, the solution time of them grows very quickly as the size of the instance increases. In this paper, we propose a binary particle swarm optimization (FBPSO) for solving the MWDSP approximately. Based on the characteristic of MWDSP, this approach designs a new position updating rule to guide the search to a promising area. An iterated greedy tabu search is used to enhance the solution quality quickly. In addition, several stochastic strategies are employed to diversify the search and prevent premature convergence. These methods maintain a good balance between the exploration and the exploitation. Experimental studies on 106 groups of 1 060 instances show that FBPSO is able to identify near optimal solutions in a short running time. The average deviation between the solutions obtained by FBPSO and the best known solutions is 0.441%. Moreover, the average solution time of FBPSO is much less than that of other existing algorithms. In particular, with the increasing of instance size, the solution time of FBPSO grows much more slowly than that of other existing algorithms.

An order-based algorithm for minimum dominating set with application in graph mining

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLSo) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLSo performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLSo that is suitable for solving the minimum weight dominating set problem. The application of RLSo in graph mining is also briefly demonstrated.

Local Search for Minimum Weight Dominating Set with Two-Level Configuration Checking and Frequency Based Scoring Function

The Minimum Weight Dominating Set (MWDS) problem is an important generalization of the Minimum Dominating Set (MDS) problem with extensive applications. This paper proposes a new local search algorithm for the MWDS problem, which is based on two new ideas. The first idea is a heuristic called two-level configuration checking (CC2), which is a new variant of a recent powerful configuration checking strategy (CC) for effectively avoiding the recent search paths. The second idea is a novel scoring function based on the frequency of being uncovered of vertices. Our algorithm is called CC2FS, according to the names of the two ideas. The experimental results show that, CC2FS performs much better than some state-of-the-art algorithms in terms of solution quality on a broad range of MWDS benchmarks.

An Effective Hybrid Memetic Algorithm for the Minimum Weight Dominating Set Problem

The minimum weight-dominating set (MWDS) problem is NP-hard and has a lot of applications in the real world. Several metaheuristic methods have been developed for solving the problem effectively, but suffering from high CPU time on large-scale instances. In this paper, we design an effective hybrid memetic algorithm (HMA) for the MWDS problem. First, the MWDS problem is formulated as a constrained 0-1 programming problem and is converted to an equivalent unconstrained 0-1 problem using an adaptive penalty function. Then, we develop a memetic algorithm for the resulting problem, which contains a greedy randomized adaptive construction procedure, a tabu local search procedure, a crossover operator, a population-updating method, and a path-relinking procedure. These strategies make a good tradeoff between intensification and diversification. A number of experiments were carried out on three types of instances from the literature. Compared with existing algorithms, HMA is able to find high-quality solutions in much less CPU time. Specifically, HMA is at least six times faster than existing algorithms on the tested instances. With increasing instance size, the CPU time required by HMA increases much more slowly than required by existing algorithms.

A hybrid self-adaptive evolutionary algorithm for the minimum weight dominating set problem

Given an undirected graph with weights associated with its vertices, the minimum weight dominating set problem MWDSP is to determinate a subset of vertices with minimum sum of weights such that each vertex of the graph either belongs to the subset or is adjacent to a vertex in the subset. This paper presents a hybrid self-adaptive evolutionary algorithm for the MWDSP. A greedy randomised adaptive construction procedure is used to generate an initial population. Then, a crossover operator and an adaptive mutation operator are employed to generate a new solution, which is further refined by an adaptive tabu search. These adaptive strategies make a good balance between intensification and diversification. Computational experiments on two types of benchmark instances prove that our proposed algorithm can find high quality solutions in a short computing time.

A hybrid algorithmic model for the minimum weight dominating set problem

Iterated greedy algorithms belong to the class of stochastic local search methods. They are based on the simple and effective principle of generating a sequence of solutions by iterating over a constructive greedy heuristic using destruction and construction phases. This paper, first, presents an efficient randomized iterated greedy approach for the minimum weight dominating set problem, where—given a vertex-weighted graph—the goal is to identify a subset of the graphs’ vertices with minimum total weight such that each vertex of the graph is either in the subset or has a neighbor in the subset. Our proposed approach works on a population of solutions rather than on a single one. Moreover, it is based on a fast randomized construction procedure making use of two different greedy heuristics. Secondly, we present a hybrid algorithmic model in which the proposed iterated greedy algorithm is combined with the mathematical programming solver CPLEX. In particular, we improve the best solution provided by the iterated greedy algorithm with the solution polishing feature of CPLEX. The simulation results obtained on a widely used set of benchmark instances shows that our proposed algorithms outperform current state-of-the-art approaches.

A hybrid evolutionary algorithm with guided mutation for minimum weight dominating set

This paper presents a hybrid evolutionary algorithm with guided mutation (EA/G) to solve the minimum weight dominating set problem (MWDS) which is Â?Â?Â?Â?$\mathcal {N}\mathcal {P}$-hard in nature not only for general graphs, but also for unit disk graphs (UDG). MWDS finds practical applications in diverse domains such as clustering in wireless networks, intrusion detection in adhoc networks, multi-document summarization in information retrieval, query selection in web databases etc. EA/G is a recently proposed evolutionary algorithm that tries to overcome the shortcomings of genetic algorithms (GAs) and estimation of distribution algorithms (EDAs) both, and that can be considered as a cross between the two. The solution obtained through EA/G algorithm is further improved through an improvement operator. We have compared the performance of our hybrid evolutionary approach with the state-of-the-art approaches on general graphs as well as on UDG. Computational results show the superiority of our approach in terms of solution quality as well as execution time.

The Minimum Weight Dominating Set Problem under Hybrid Uncertain Environments

The minimum weight dominating set problem (MWDSP) has been a popular research topic in recent years. The weights of vertexes may be considered as cost, time, or opponent’s payoff, which are uncertain in most cases. This paper discusses MWDSP under hybrid uncertain environments where the weights of vertexes are random fuzzy variables. First, random fuzzy theory is introduced to describe these hybrid uncertain variables. Then we propose three decision models based on three different decision criteria to solve MWDSP under hybrid uncertain environments. To solve the proposed models, we present a hybrid intelligent algorithm where random fuzzy simulation and genetic algorithm are embedded. Numerical experiments are performed in the last to show the robustness and effectiveness of the presented hybrid intelligent algorithm.

Hybrid metaheuristic algorithms for minimum weight dominating set

Minimum weight dominating set (MWDS) finds many uses in solving problems as varied as clustering in wireless networks, multi-document summarization in information retrieval and so on. It is proven to be NP-hard, even for unit disk graphs. Many centralized and distributed, greedy and approximation algorithms have been proposed for the MWDS problem. However, all the approximation algorithms are limited to unit disk graphs which are primarily used to model wireless networks. This assumption fails when applied to other domains. In this paper, we present two metaheuristic algorithms – a hybrid genetic algorithm and a hybrid ant colony optimization algorithm – for the problem of computing minimum weight dominating set. We compare our results with that of a greedy heuristic as well as the only other metaheuristic algorithm proposed so far in the literature and show that our algorithms are far better than these algorithms.

The set covering problem on circulant matrices: polynomial instances and the relation with the dominating set problem on webs

Abstract The dominating set polyhedron of a web graph W n k is the set covering polyhedron of a circulant matrix C n 2 k + 1 . In a previous work we generalize the results by Argiroffo and Bianchi on valid inequalities associated with every circulant minor of a circulant matrix and we conjecture that, for any k, the minor inequalities together with the boolean facets and the rank constraint are enough to describe the set covering polyhedron of C n k . In this work we prove that the conjecture is true for the family of C s k + r k with s = 2 , 3 and 0 ⩽ r ⩽ s − 1 and give a polynomial separation algorithm for inequalities involved in the description. Thus, we prove the polynomiality of the set covering problem on these families. As a consequence we obtain the polynomiality of the minimum weight dominating set problem on webs of the form W n t , when n = 2 s t + s + r with s = 2 , 3 and 0 ⩽ r ⩽ s − 1 .

Minimum average routing path clustering problem in multi-hop 2-D underwater sensor networks

In this paper, we introduce a new clustering problem in underwater sensor networks, namely minimum average routing path clustering problem (MARPCP). To deal with the high complexity of MARPCP, we relax it to a special case of minimum weight dominating set problem (MWDSP). We show an existing algorithm for MWDSP can produce an approximate solution for MARPCP. Also, we design a constant factor approximation algorithm for MARPCP, which is much faster than the first method.

On the dominating set polytope of web graphs

Abstract The dominating set polyhedron of a web graph W n k is the set covering polyhedron of a circulant matrix C n 2 k + 1 . Working on the set covering polyhedron of general circulant matrices, Argiroffo and Bianchi found a class of valid inequalities, induced by a particular family of circulant minors. The authors also identified sufficient conditions on the minors in order to induce facet defining inequalities. In this work we generalize these results for new valid inequalities associated with every circulant minor. We conjecture that, for any k, the minor inequalities together with the boolean facets are enough to describe the set covering polyhedron of C n k . This conjecture is true for k = 3 as Bouchakour et al. prove working in the context of the Minimum Weight Dominating Set problem (MWDSP) on cycles, i.e, on webs W n 1 . They also derive the polynomiality of MWDSP on cycles by giving a polynomial separation algorithm for minor inequalities of C n 3 . Although the computational complexity of the separation problem of every minor inequality is still an open problem, we can state that the set covering problem on C n k matrices is polynomial for a fixed k and obtain polynomial separation algorithms for particular classes of minor inequalities.

A BETTER APPROXIMATION FOR MINIMUM AVERAGE ROUTING PATH CLUSTERING PROBLEM IN 2-D UNDERWATER SENSOR NETWORKS

Previously, we proposed Minimum Average Routing Path Clustering Problem (MARPCP) in multi-hop USNs. The goal of this problem is to find a clustering of a USN so that the average clustering-based routing path from a node to it nearest underwater sink is minimized. We relaxed MARPCP to a special case of Minimum Weight Dominating Set Problem (MWDSP), namely MWDSP-R. In addition, we showed the Performance Ratio (PR) of α-approximation algorithm for MWDSP-R is 3α for MARPCP. Based on this result, we showed the existence of a (15 + ∊)-approximation algorithm for MARPCP. In this paper, we first establish the NP-completeness of both MARPCP and MWDSP-R. Then, we propose a PTAS for MWDSP-R. By combining this result with our previous one, we have a (3 + ∊)-approximation algorithm for MARPCP.

A [formula omitted]-approximation algorithm for minimum weighted dominating set in unit disk graph

We study the minimum weight dominating set problem in weighted unit disk graph, and give a polynomial time algorithm with approximation ratio 5 + ϵ , improving the previous best result of 6 + ϵ in [Yaochun Huang, Xiaofeng Gao, Zhao Zhang, Weili Wu, A better constant-factor approximation for weighted dominating set in unit disk graph, J. Comb. Optim. (ISSN: 1382-6905) (2008) 1573–2886. (Print) (Online)]. Combining the common technique used in the above mentioned reference, we can compute a minimum weight connected dominating set with approximation ratio 9 + ϵ , beating the previous best result of 10 + ϵ in the same work.