The Minimum Vertex Cover (MVC) problem is a well-known combinatorial optimization problem of great importance in theory and applications. In recent years, local search has been shown to be an effective and promising approach to solve hard problems, such as MVC. In this paper, we introduce two new local search algorithms for MVC, called EWLS (Edge Weighting Local Search) and EWCC (Edge Weighting Configuration Checking). The first algorithm EWLS is an iterated local search algorithm that works with a partial vertex cover, and utilizes an edge weighting scheme which updates edge weights when getting stuck in local optima. Nevertheless, EWLS has an instance-dependent parameter. Further, we propose a strategy called Configuration Checking for handling the cycling problem in local search. This is used in designing a more efficient algorithm that has no instance-dependent parameters, which is referred to as EWCC. Unlike previous vertex-based heuristics, the configuration checking strategy considers the induced subgraph configurations when selecting a vertex to add into the current candidate solution. A detailed experimental study is carried out using the well-known DIMACS and BHOSLIB benchmarks. The experimental results conclude that EWLS and EWCC are largely competitive on DIMACS benchmarks, where they outperform other current best heuristic algorithms on most hard instances, and dominate on the hard random BHOSLIB benchmarks. Moreover, EWCC makes a significant improvement over EWLS, while both EWLS and EWCC set a new record on a twenty-year challenge instance. Further, EWCC performs quite well even on structured instances in comparison to the best exact algorithm we know. We also study the run-time behavior of EWLS and EWCC which shows interesting properties of both algorithms.