On the Hardness of Scheduling With Non-Uniform Communication Delays

TL;DR

This paper proves that scheduling with non-uniform communication delays cannot be approximated within any constant factor, establishing a logarithmic hardness result under standard complexity assumptions, and introduces the UMPS problem as central to this hardness.

Contribution

It demonstrates a logarithmic hardness of approximation for scheduling with non-uniform communication delays and introduces the UMPS problem as key to understanding this complexity.

Findings

Logarithmic hardness of approximation proven for the problem.

Simple reduction from the UMPS problem to establish hardness.

Conjecture that UMPS is very hard to approximate, implying broader hardness results.

Abstract

In the scheduling with non-uniform communication delay problem, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. One of the top ten open problems in scheduling theory, in the influential list by Schuurman and Woeginger and its latest update by Bansal, asks if the problem admits a constant factor approximation algorithm. In this paper, we answer the question in negative by proving that there is a logarithmic hardness for the problem under the standard complexity theory assumption that NP-complete problems do not admit quasi-polynomial time algorithms. Our hardness result is obtained using a surprisingly simple reduction from a problem that we call Unique Machine Precedence constraints Scheduling (UMPS). We believe that this problem is of central importance in understanding the hardness of many scheduling problems and conjecture that it is very hard to approximate. Among other things, our conjecture implies a logarithmic hardness of related machine scheduling with precedences, a long-standing open problem in scheduling theory and approximation algorithms.