To address the limited solving accuracy of the existing heuristic algorithms for Minimum Weakly Connected Dominating Set Problem （MWCDSP）， a Master-Apprentice Evolutionary algorithm based on improved Local Search （MAE-LS） was proposed. First， double simplification rules were used in the initial solution construction stage to identify vertices that must be included in the optimal solution， thereby narrowing the searching space and improving quality of the initial solution. Second， a feasible solution construction mechanism with priority of weak connectivity was introduced in the local search stage. Compared to traditional methods， this mechanism maintained the weak connectivity and dominance of solutions more effectively. At the same time， combined with an efficient vertex selection strategy， the addition to the candidate solutions or removal from the solutions of vertices was guided dynamically， thereby improving diversity and feasibility of the solutions. Third， a dual cycle suppression mechanism combining frequency and tabu strategy was designed to effectively reduce the probability of repeated searches. Finally， a path-breaking strategy and a perturbation strategy were proposed to help the algorithm escape from the local optimum. The proposed algorithm was compared with the CPLEX （C Programming Language for EXpressions） exact solver and 6 state-of-the-art algorithms on 98 instances of the 4 benchmark test sets. Experimental results show that MAE-LS is the algorithm obtaining the largest number of optimal solutions， especially on the NDR （Network Data Repository） instance， where the number of optimal solutions obtained by MAE-LS is 61.5% higher than that of the algorithm with suboptimal performance， FPLS （Local Search algorithm based on the Feedback mechanism and the Perturbation strategy）， significantly， fully proving the significant advantages of this algorithm in terms of solving accuracy and algorithm stability. MAE-LS provides an effective solution to NP （Nondeterministic Polynomial） -hard network optimization problems， and is of great value to practical applications such as wireless sensor network construction.