To date, the only way to argue polynomial lower bounds for dynamic algorithms is via fine-grained complexity arguments. These arguments rely on strong assumptions about specific problems such as the Strong Exponential Time Hypothesis (SETH) and the Online Matrix-Vector Multiplication Conjecture (OMv). While they have led to many exciting discoveries, dynamic algorithms still miss out some benefits and lessons from the traditional ``coarse-grained'' approach that relates together classes of problems such as P and NP. In this paper we initiate the study of coarse-grained complexity theory for dynamic algorithms. Below are among questions that this theory can answer. What if dynamic Orthogonal Vector (OV) is easy in the cell-probe model? A research program for proving polynomial unconditional lower bounds for dynamic OV in the cell-probe model is motivated by the fact that many conditional lower bounds can be shown via reductions from the dynamic OV problem. Since the cell-probe model is more powerful than word RAM and has historically allowed smaller upper bounds, it might turn out that dynamic OV is easy in the cell-probe model, making this research direction infeasible. Our theory implies that if this is the case, there will be very interesting algorithmic consequences: If dynamic OV can be maintained in polylogarithmic worst-case update time in the cell-probe model, then so are several important dynamic problems such as $k$-edge connectivity, $(1+\epsilon)$-approximate mincut, $(1+\epsilon)$-approximate matching, planar nearest neighbors, Chan's subset union and 3-vs-4 diameter. The same conclusion can be made when we replace dynamic OV by, e.g., subgraph connectivity, single source reachability, Chan's subset union, and 3-vs-4 diameter. Lower bounds for $k$-edge connectivity via dynamic OV? (see the full abstract in the pdf file).

翻译：迄今为止，对动态算法进行多项式下界论证的唯一途径是通过细粒度复杂性论证。这些论证依赖于对特定问题的强假设，例如强指数时间假设（SETH）和在线矩阵向量乘法猜想（OMv）。尽管这些方法带来了许多激动人心的发现，但动态算法仍未充分受益于传统“粗粒度”方法（例如将P和NP等问题类别关联起来）的优势与经验教训。本文首次系统研究动态算法的粗粒度复杂性理论。该理论可解答以下问题：若动态正交向量问题（Orthogonal Vector, OV）在单元探针模型（cell-probe model）中易于求解会怎样？由于许多条件性下界可通过动态OV问题的归约得到证明，一个旨在为动态OV在单元探针模型中建立无条件多项式下界的研究计划应运而生。然而，单元探针模型比字RAM（word RAM）更强大且历史上允许更小的上界，因此动态OV可能在该模型中易于求解，导致该研究方向不可行。我们的理论表明：若此情况成立，将产生极其重要的算法推论——若动态OV可在单元探针模型中实现多对数最坏情况更新时间的维护，则诸多重要动态问题（如$k$边连通性、$(1+\epsilon)$近似最小割、$(1+\epsilon)$近似匹配、平面最近邻、Chan子集并问题以及3-vs-4直径问题）同样可在该模型中实现。若将动态OV替换为子图连通性、单源可达性、Chan子集并问题或3-vs-4直径问题，亦可得出相同结论。动态OV能否为$k$边连通性问题提供下界？（完整摘要见PDF文件）