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Communication performance is critical for IoT applications. According to previous research, an internet-based overlay model is feasible for the implementation of the IoT. One important issue in the overlay routing model is the overlay node placement problem (ONPP). Once the size of overlay node set is fixed to a particular number k, the ONPP changes to k-ONPP. In this work, the IoT-based overlay node placement problem is formulized and analyzed. The major contributions of the paper include providing the time complexity of multi hop k-ONPP and its theoretical limit boundary of approximation ratio and proposing a local search algorithm. Furthermore, the time complexity and approximation ratio boundary of the local search algorithm are given. The proposed local search algorithm is evaluated by both time and efficiency where efficiency refers to the degree of approximation of algorithm results with optimal solutions. Another algorithm, TAG, is used for comparison. Finally, a simulation experiment based on network simulator EstiNet is provided. 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