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Community search is important in social network analysis. For a given vertex in a graph, the goal is to find the best community the vertex belongs to. Intuitively, the best community for a given vertex should be in the vicinity of the vertex. However, existing solutions use \emph{global search} to find the best community. These algorithms, although straight-forward, are very costly, as all vertices in the graph may need to be visited. In this paper, we propose a \emph{local search} strategy, which searches in the neighborhood of a vertex to find the best community for the vertex. We show that, because the minimum degree measure used to evaluate the goodness of a community is not \emph{monotonic}, designing efficient local search solutions is a very challenging task. We present theories and algorithms of local search to address this challenge. The efficiency of our local search strategy is verified by extensive experiments …