## 2 Related Works  
Finding the dominant sets in a graph is closely related to understanding the nature of the data or structure to which it is applied. The complexity of the problem and the exponential solution encourage the continuous discovery of innovative strategies that work faster. Greedy algorithms, in particular, have provided remarkable solutions to this NP-hard complexity. This section reviews recent developments in the field and presents the variety of approaches and the ongoing development of algorithms that address the MDS problem. [PERSON] and [PERSON] proposed a method that identifies important nodes and edges in a given graph and thus reveals shortcuts in the graph [24]. Their proposed method has been tested on real-world datasets and its effectiveness has been demonstrated in network analysis. Following this paper, [PERSON] proposed a new order-based random local search (RLS\({}_{0}\)) algorithm to solve the MDS problem [17]. Experimental results show that the RLS\({}_{0}\) algorithm outperforms both the greedy algorithm and two ACO-based (ant colony optimization) metaheuristic algorithms and is also scalable to large graphs. [PERSON] et al. proposed FastMWDS, a fast local search method, to tackle the MDS problem [25], especially for large-scale graphs. This algorithm handles large-scale networks effectively by using heuristic techniques. When compared to other heuristic approaches, the study showed better performance both in solution quality and execution time. [PERSON] et al. introduced an efficient local search algorithm to solve the MDS problem in large graphs [26]. This algorithm, named Score Checking and Best-picking with Probabilistic Walk (ScBppw), utilizes a probabilistic approach combined with a Tabu Search strategy to diversify the search process. [PERSON] et al. introduced an efficient local search algorithm named FastDS to address the MDS problem [27]. This algorithm integrates two main ideas: a novel local search framework and a construction procedure with inference rules. Notably, FastDS has been reported to achieve significantly better solutions compared to state-of-the-art algorithms on large graphs. The time complexity of FastDS is \(O(nlogn+m)\), where n represents the number of vertices and m represents the number of edges. [PERSON] et al. have designed a unified \(O(ln\delta)\)-approximation algorithm for the generalized MDS problem, with the concept of submodular and weakly submodular potential functions [28]. The authors have stated that this algorithm either provides the best result among the existing approximation algorithms for specific versions or presents the first approximate solution. In their research, [PERSON] et al. proposed a local search algorithm for the minimum positive influence dominating set (MPIDS) problem that includes reduction-based initialization, a new node exchange strategy, and a general local search framework [29]. This method shows improvement over previous MPIDS solutions on various graph benchmarks, presenting a new way to tackle NP-hard combinatorial optimization problems in graph theory. [PERSON] introduced a method to pinpoint nodes that form the MDS in graphs with a minimum degree of three, transforming these graphs into 2-connected forms with specific cycle lengths [30]. This approach, which cleverly labels nodes within these cycles, offers a partial solution to the APX-complete MDS problem for cubic graphs, contributing to the field of graph theory with a novel perspective on combinatorial optimization challenges. In another study, [PERSON] et al. tackle the NP-hard MDS problem by proposing an Iterated Greedy algorithm, showcasing its effectiveness in finding optimum or nearly optimum solutions through a comparative study with exact and up-to-date algorithms [31]. Their algorithm achieves an average deviation of only 0.04% from optimal solutions for solvable instances and 1.23% for more complex cases, thus presenting a significant advancement in solving combinatorial optimization problems in graph theory. [PERSON] et al.