A recent paper by Abboud and Wallheimer [ITCS 2023] presents self-reductions for various fundamental graph problems, that transform worst-case instances to expanders, thus proving that the complexity remains unchanged if the input is assumed to be an expander. An interesting corollary of their self-reductions is that, if some problem admit such reduction, then the popular algorithmic paradigm based on expander-decompositions is useless against it. In this paper, we improve their core gadget, which augments a graph to make it an expander while retaining its important structure. Our new core construction has the benefit of being simple to analyze and generalize, while obtaining the following results: 1. A derandomization of the self-reductions, showing that the equivalence between worst-case and expander-case holds even for deterministic algorithms, and ruling out the use of expander-decompositions as a derandomization tool. 2. An extension of the results to other models of computation, such as the Fully Dynamic model and the Congested Clique model. In the former, we either improve or provide an alternative approach to some recent hardness results for dynamic expander graphs, by Henzinger, Paz, and Sricharan [ESA 2022]. In addition, we continue this line of research by designing new self-reductions for more problems, such as Max-Cut and dynamic Densest Subgraph, and demonstrating that the core gadget can be utilized to lift lower bounds based on the OMv Conjecture to expanders.

翻译：Abboud和Wallheimer的最新论文[ITCS 2023]针对多种基础图问题提出了自归约方法，将最坏情形实例转化为扩张图，从而证明假设输入为扩张图时问题复杂性保持不变。这些自归约的有趣推论是：若某问题容许此类归约，则基于扩张图分解的流行算法范式对其无效。本文改进了其核心构件，该构件能在保留图重要结构的同时将其增强为扩张图。新核心构造具有易于分析和推广的优势，并取得以下成果：1. 实现自归约的去随机化，证明最坏情形与扩张图情形之间的等价性对确定性算法同样成立，并排除使用扩张图分解作为去随机化工具的可能性。2. 将结果扩展至其他计算模型，如全动态模型和拥塞团模型。在前一模型中，我们改进或为Henzinger、Paz和Sricharan [ESA 2022]关于动态扩张图的最新困难性结果提供替代方案。此外，我们延续该研究方向，为更多问题（如最大割和动态最密子图）设计新的自归约，并展示核心构件可用于将基于OMv猜想的理论下界提升至扩张图。