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

## TL;DR

This paper investigates the computational complexity of minimizing communication costs in a weighted network model for parallel computation, establishing hardness results for various problem variants.

## Contribution

It introduces the Weighted Massively Parallel Computation (WMPC) model and characterizes the complexity of related communication minimization problems.

## Key findings

- Some problems are in P
- Some are NP-complete
- Some are W[1]-complete

## Abstract

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.