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

The Hardness of Optimization Problems on the Weighted Massively Parallel Computation Model