  The topology-aware Massively Parallel Computation (MPC) model is proposed and
studied recently, which enhances the classical MPC model by the awareness of
network topology. The work of Hu et al. on topology-aware MPC model considers
only the tree topology. In this paper a more general case is considered, where
the underlying network is a weighted complete graph. We then call this model as
Weighted Massively Parallel Computation (WMPC) model, and study the problem of
minimizing communication cost under it. Two communication cost minimization
problems are defined based on different pattern of communication, which are the
Data Redistribution Problem and Data Allocation Problem. We also define four
kinds of objective functions for communication cost, which consider the total
cost, bottleneck cost, maximum of send and receive cost, and summation of send
and receive cost, respectively. Combining the two problems in different
communication pattern with the four kinds of objective cost functions, 8
problems are obtained. The hardness results of the 8 problems make up the
content of this paper. With rigorous proof, we prove that some of the 8
problems are in P, some FPT, some NP-complete, and some W[1]-complete.
