Scheduling Dags under Uncertainty

TL;DR

This paper studies a parallel scheduling problem for DAGs on unreliable workers, providing polynomial algorithms under certain restrictions and proving NP-hardness and inapproximability results in more general cases.

Contribution

It introduces a new scheduling model with probabilistic worker reliability, offers a polynomial solution for restricted cases, and establishes complexity bounds for general scenarios.

Findings

Polynomial algorithm for fixed dag width and worker count

NP-hardness when either dag width or worker count grows

Inapproximability within 5/4 factor for general case

Abstract

This paper introduces a parallel scheduling problem where a directed acyclic graph modeling $t$ tasks and their dependencies needs to be executed on $n$ unreliable workers. Worker $i$ executes task $j$ correctly with probability $p_{i,j}$. The goal is to find a regimen $\Sigma$, that dictates how workers get assigned to tasks (possibly in parallel and redundantly) throughout execution, so as to minimize the expected completion time. This fundamental parallel scheduling problem arises in grid computing and project management fields, and has several applications. We show a polynomial time algorithm for the problem restricted to the case when dag width is at most a constant and the number of workers is also at most a constant. These two restrictions may appear to be too severe. However, they are fundamentally required. Specifically, we demonstrate that the problem is NP-hard with constant number of workers when dag width can grow, and is also NP-hard with constant dag width when the number of workers can grow. When both dag width and the number of workers are unconstrained, then the problem is inapproximable within factor less than 5/4, unless P=NP.