This paper investigates network partition and edge server placement problem to exploit the benefit of edge computing for distributed state estimation. A constrained many-objective optimization problem is formulated to minimize the cost of edge server deployment, operation, and maintenance, avoid the difference in the partition sizes, reduce the level of coupling between connected partitions, and maximize the inner cohesion of each partition. Capacities of edge server are constrained against underload and overload. To efficiently solve the problem, an improved non-dominated sorting genetic algorithm III (NSGA-III) is developed, with a specifically designed directed mutation operator based on topological characteristics of the partitions to accelerate convergence. Case study validates that the proposed formulations effectively characterize the practical concerns and reveal their trade-offs, and the improved algorithm outperforms existing representative ones for large-scale networks in converging to a near-optimal solution. The optimized result contributes significantly to real-time distributed state estimation.