A recent paper by Abboud and Wallheimer [ITCS 2023] presents self-reductions for various fundamental graph problems, which 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 admits 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最近的一篇论文[ITCS 2023]提出了多种基础图问题的自归约方法，这些方法将最坏情况实例转换为扩展图，从而证明即使假设输入为扩展图，问题的计算复杂度保持不变。他们自归约的一个有趣推论是：若某个问题允许此类归约，则基于扩展图分解的主流算法范式对该问题无效。本文改进了他们的核心构件——该构件通过增强图结构使其成为扩展图，同时保留其关键性质。我们提出的新核心构造具有易于分析和推广的优点，并取得了以下成果：1. 实现了自归约的去随机化，证明最坏情况与扩展图情况的等价性对确定性算法依然成立，从而排除了将扩展图分解作为去随机化工具的可能性。2. 将结果推广至其他计算模型，如完全动态模型和拥塞团模型。对于前者，我们改进了Henzinger、Paz和Sricharan[ESA 2022]近期关于动态扩展图硬度证明的部分结果，或提供了替代证明方法。此外，我们通过为更多问题（如最大割问题和动态最稠密子图问题）设计新的自归约，延续了这一研究方向，并证明该核心构件可将基于OMv猜想的下界结果提升至扩展图场景。