Graph matching plays an essential role in many real applications. In this paper, we study how to match two large graphs by maximizing the number of matched edges, which is known as maximum common subgraph matching and is NP-hard. To find exact matching, it cannot a graph with more than 30 nodes. To find an approximate matching, the quality can be very poor. We propose a novel two-step approach that can efficiently match two large graphs over thousands of nodes with high matching quality. In the first step, we propose an anchor-selection/expansion approach to compute a good initial matching. In the second step, we propose a new approach to refine the initial matching. We give the optimality of our refinement and discuss how to randomly refine the matching with different combinations. We further show how to extend our solution to handle labeled graphs. We conducted extensive testing using real and synthetic datasets and report our findings in this paper.