We build on the work of Berg, Boyar, Favrholdt, and Larsen, who developed a complexity theory for online problems with and without predictions (IJTCS-FAW, volume 15828 of LNCS, Springer, 2025) where they define a hierarchy of complexity classes that classifies online problems based on the competitiveness of best possible deterministic online algorithms for each problem. Their work focused on online minimization problems and we continue their work by considering online maximization problems. We compare the competitiveness of the base online minimization problem from Berg, Boyar, Favrholdt, and Larsen, Asymmetric String Guessing, to the competitiveness of Online Bounded Degree Independent Set. Formally, we show that there exists algorithms of any given competitiveness for Asymmetric String Guessing if and only if there exists algorithms of the same competitiveness for Online Bounded Degree Independent Set, while respecting that the competitiveness of algorithms is measured differently for minimization and maximization problems. Beyond this, we give several hardness preserving reductions between different online maximization problems, which imply new membership, hardness, and completeness results for the complexity classes. Finally, we show new positive and negative algorithmic results for (among others) Online Bounded Degree Independent Set, Online Interval Scheduling, Online Set Packing, and Online Bounded Degree Clique.

翻译：本文基于Berg、Boyar、Favrholdt和Larsen的研究工作展开（见IJTCS-FAW，LNCS第15828卷，Springer，2025），该研究构建了含预测与不含预测在线问题的计算复杂性理论，通过最优确定性在线算法的竞争比定义了在线问题的复杂性层次结构。原研究聚焦于在线最小化问题，本文则延续其工作，系统考察在线最大化问题。我们将Berg等人提出的基础在线最小化问题——非对称字符串猜测问题——与在线有界度独立集问题的竞争比进行形式化比较。具体而言，我们证明：对于任意给定的竞争比，存在具有该竞争比的非对称字符串猜测算法，当且仅当存在具有相同竞争比的在线有界度独立集算法（需注意最小化与最大化问题的竞争比度量方式存在差异）。此外，我们在多个在线最大化问题之间建立了保持计算难度的归约关系，这些归约推导出复杂性类别中新的成员归属、困难性及完备性结果。最后，我们针对在线有界度独立集、在线区间调度、在线集合覆盖及在线有界度团等问题，提出了新的正向算法结果与负向不可行性结论。