Online vector scheduling and generalized load balancing

We give a polynomial time reduction from the vector scheduling problem (VS) to the 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 _Lln(md)$L_{ln\left(md\right)}$ norm of the resulting load when it comes, where m$m$ is the number of partitions and d$d$ is the dimension of vectors. It has an approximation bound of elog(md)$elog\left(md\right)$, which is in O(ln(md))$O(ln\left(md\right))$, so it also improves the O(ln²d)$O\left({ln}^{2}d\right)$ 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$P=NP$.