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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. |
