We study multidimensional resource scheduling. Each job $i$ has a $d$-dimensional resource-demand vector $v_i$ and a processing time $s_i$. The scheduler assigns a start time to each job, subject to the constraint that, at every time, the total demand of the jobs being processed does not exceed $1$ in any resource dimension. The objective is to minimize the makespan. We focus on the regime in which every individual resource demand is small. We ask whether the favorable \emph{small-vector phenomenon} known for multidimensional vector packing, which corresponds to the special case of unit processing times, extends to jobs with heterogeneous processing times. The key difficulty is that arbitrary processing times create temporal interactions across multiple duration scales: a long job may overlap many shorter jobs, while feasibility must be maintained throughout every job's execution interval. We prove that the small-vector phenomenon persists in this temporal setting. For any $0<ε<1/4$, after normalizing the maximum processing time to $1$, if every coordinate of every demand vector is at most $O(ε^2/\log(d/ε))$, we give a randomized offline algorithm that produces a schedule with expected makespan at most $ (1+6ε)\mathrm{OPT}+3$. Thus, sufficiently small resource demands admit asymptotically near-optimal schedules in arbitrary dimension, despite heterogeneous processing times. We also obtain constant competitive ratios in the online setting. Let $T$ denote the ratio between the maximum and minimum processing times. If every coordinate is at most $O(1/(\log d\log T))$, we give a randomized $O(1)$-competitive algorithm, with a competitive ratio independent of both $d$ and $T$. We further derandomize our approach, obtaining a deterministic $O(1)$-competitive algorithm under a comparable smallness assumption.

翻译：暂无翻译