Given an attributed graph$G$and a query node$q$, Community Search over Attributed Graphs (CS-AG) aims to find a structure- and attribute-cohesive subgraph from$G$that contains$q$. Although CS-AG has been widely studied, they still face three challenges. (1) Exact methods based on graph traversal are time-consuming, especially for large graphs. Some tailored indices can improve efficiency, but introduce nonnegligible storage and maintenance overhead. (2) Approximate methods with a loose approximation ratio only provide a coarse-grained evaluation of a community's quality, rather than a reliable evaluation with an accuracy guarantee in runtime. (3) Attribute cohesiveness metrics often ignores the important correlation with the query node$q$. We formally define our CS-AG problem atop a$q- \mathbf{centric}$attribute cohesiveness metric considering both textual and numerical attributes, for$k-\mathbf{core}$model on homogeneous graphs. We show the problem is NP-hard. To solve it, we first propose an exact baseline with three pruning strategies. Then, we propose an index-free sampling-estimation-based method to quickly return an approximate community with an accuracy guarantee, in the form of a confidence interval. Once a good result satisfying a user-desired error bound is reached, we terminate it early. We extend it to heterogeneous graphs,$k-\mathbf{truss}$model, and size-bounded CS. Comprehensive experimental studies on ten real-world datasets show its superiority, e.g., at least$1.54\times (41.1\times$on average) faster in response time and a reliable relative error (within a user-specific error bound) of attribute cohesiveness is achieved.