Sublinear Approximation Schemes for Scheduling Precedence Graphs of Bounded Depth

TL;DR

This paper introduces the first sublinear space and time approximation schemes for scheduling precedence graphs of bounded depth, achieving near-optimal solutions under specific constraints in big data environments.

Contribution

It presents novel one-pass streaming and sublinear time algorithms for scheduling with precedence constraints, extending to more general job size scenarios.

Findings

First sublinear space approximation scheme for bounded depth scheduling.

First sublinear time algorithms for the same scheduling problem.

Achieves near-optimal approximation ratios under certain constraints.

Abstract

We study the classical scheduling problem on parallel machines %with precedence constraints where the precedence graph has the bounded depth $h$. Our goal is to minimize the maximum completion time. We focus on developing approximation algorithms that use only sublinear space or sublinear time. We develop the first one-pass streaming approximation schemes using sublinear space when all jobs' processing times differ no more than a constant factor $c$ and the number of machines $m$ is at most $\tfrac {2n \epsilon}{3 h c }$. This is so far the best approximation we can have in terms of $m$, since no polynomial time approximation better than $\tfrac{4}{3}$ exists when $m = \tfrac{n}{3}$ unless P=NP. %the problem cannot be approximated within a factor of $\tfrac{4}{3}$ when $m = \tfrac{n}{3}$ even if all jobs have equal processing time. The algorithms are then extended to the more general problem where the largest $\alpha n$ jobs have no more than $c$ factor difference. % for some constant $0 < \alpha \le 1$. We also develop the first sublinear time algorithms for both problems. For the more general problem, when $ m \le \tfrac { \alpha n \epsilon}{20 c^2 \cdot h } $, our algorithm is a randomized $(1+\epsilon)$-approximation scheme that runs in sublinear time. This work not only provides an algorithmic solution to the studied problem under big data % and cloud computing environment, but also gives a methodological framework for designing sublinear approximation algorithms for other scheduling problems.