In this paper, we propose two new strategies to design efficient local search algorithms for the minimum vertex cover (MVC) problem. There are two main drawbacks in state-of-the-art MVC local search algorithms: First, they select a pair of vertices to be exchanged simultaneously, which is time consuming; Second, although they use edge weighting techniques, they do not have a strategy to decrease the weights. To address these drawbacks, we propose two new strategies: two stage exchange and edge weighting with forgetting. The two stage exchange strategy selects two vertices to be exchanged separately and performs the exchange in two stages. The strategy of edge weighting with forgetting not only increases weights of uncovered edges, but also decreases some weights for each edge periodically. We utilize these two strategies to design a new algorithm dubbed NuMVC. The experimental results show that NuMVC significantly outperforms existing state-of-the-art heuristic algorithms on most of the hard DIMACS instances and all instances in the hard random BHOSLIB benchmark.