We consider the problem of network completion, which is to make the minimum amount of modifications to a given network so that the resulting network is most consistent with the observed data. We employ here a certain type of differential equations as gene regulation rules in a genetic network, gene expression time series data as observed data, and deletions and additions of edges as basic modification operations. In addition, we assume that the numbers of deleted and added edges are specified. For this problem, we present a novel method using dynamic programming and least-squares fitting and show that it outputs a network with the minimum sum squared error in polynomial time if the maximum indegree of the network is bounded by a constant. We also perform computational experiments using both artificially generated and real gene expression time series data.