This paper studies the online vector bin packing (OVBP) problem and the related problem of online hypergraph coloring (OHC). Firstly, we use a double counting argument to prove an upper bound of the competitive ratio of $FirstFit$ for OVBP. Our proof is conceptually simple, and strengthens the result in Azar et. al. by removing the dependency on the bin size parameter. Secondly, we introduce a notion of an online incidence matrix that is defined for every instance of OHC. Using this notion, we provide a reduction from OHC to OVBP, which allows us to carry known lower bounds of the competitive ratio of algorithms for OHC to OVBP. Our approach significantly simplifies the previous argument from Azar et. al. that relied on using intricate graph structures. In addition, we slightly improve their lower bounds. Lastly, we establish a tight bound of the competitive ratio of algorithms for OHC, where input is restricted to be a hypertree, thus resolving a conjecture in Nagy-Gyorgy et. al. The crux of this proof lies in solving a certain combinatorial partition problem about multi-family of subsets, which might be of independent interest.

翻译：本文研究在线向量装箱（OVBP）问题及其相关问题在线超图着色（OHC）。首先，我们利用双计数论证证明了OVBP中首次适应算法（FirstFit）竞争比的上界。该证明概念简洁，通过消除对箱子尺寸参数的依赖，强化了Azar等人的结果。其次，我们引入在线关联矩阵概念，该矩阵针对每个OHC实例定义。利用这一概念，我们给出从OHC到OVBP的归约，从而可将OHC算法竞争比的已知下界迁移至OVBP。该方法显著简化了Azar等人先前依赖复杂图结构的论证，同时对其下界略有改进。最后，我们建立了输入限制为超树的OHC算法竞争比的紧界，由此解决了Nagy-Gyorgy等人提出的猜想。该证明的核心在于求解某类关于子集多族的组合划分问题，该问题本身可能具有独立研究价值。