Online Vector Scheduling and Generalized Load Balancing

TL;DR

This paper introduces a polynomial time reduction from vector scheduling to generalized load balancing, enabling a simple online algorithm with improved approximation bounds and establishing computational hardness results.

Contribution

It provides the first non-trivial online algorithm for vector scheduling and demonstrates the computational hardness of generalized load balancing.

Findings

Online algorithm minimizes L_{ln(md)} norm upon vector arrival

Approximation bound of e*log(md), improving previous bounds

Reduction shows no constant approximation for GLB unless P=NP

Abstract

We give a polynomial time reduction from vector scheduling problem (VS) to generalized load balancing problem (GLB). This reduction gives the first non-trivial online algorithm for VS where vectors come in an online fashion. The online algorithm is very simple in that each vector only needs to minimize the $L_{\ln(md)}$ norm of the resulting load when it comes, where $m$ is the number of partitions and $d$ is the dimension of vectors. It has an approximation bound of $e\log(md)$, which is in $O(\ln(md))$, so it also improves the $O(\ln^2d)$ bound of the existing polynomial time algorithm for VS. Additionally, the reduction shows that GLB does not have constant approximation algorithms that run in polynomial time unless $P=NP$.