动态算法的粗粒度复杂性
Coarse-Grained Complexity for Dynamic Algorithms
AI总结：
本文开创动态算法的粗粒度复杂性理论，研究若动态OV问题在单元探针模型中易解，将导致多个重要动态问题获得多对数时间算法，并探讨通过动态OV归约证明下界。
摘要：
迄今为止，论证动态算法的多项式下界的唯一途径是细粒度复杂性论证。这些论证依赖于对特定问题的强假设，例如强指数时间假说（SETH）和在线矩阵-向量乘法猜想（OMv）。虽然这些论证带来了许多令人兴奋的发现，但动态算法仍然缺少传统“粗粒度”方法的一些益处和教训，该方法将诸如P和NP等问题类别联系起来。在本文中，我们开创了动态算法的粗粒度复杂性理论研究。以下是该理论能够回答的一些问题。\n\n如果动态正交向量（OV）问题在单元探针模型中容易解决怎么办？一个在单元探针模型中证明动态OV问题的无条件多项式下界的研究项目，其动机在于许多条件性下界可以通过从动态OV问题的归约来展示。由于单元探针模型比字RAM更强大，并且在历史上允许更小的上界，因此动态OV问题在单元探针模型中可能变得容易，从而使这一研究方向不可行。我们的理论表明，如果情况如此，将会产生非常有趣的算法后果：如果动态OV问题可以在单元探针模型中以多对数最坏情况更新时间维护，那么几个重要的动态问题也可以，例如$k$边连通性、$(1+ε)$近似最小割、$(1+ε)$近似匹配、平面最近邻、Chan的子集并集和3对4直径。当我们用例如子图连通性、单源可达性、Chan的子集并集和3对4直径替换动态OV问题时，也可以得出相同的结论。\n\n通过动态OV问题得到$k$边连通性的下界？（完整摘要见PDF文件）。
英文摘要：
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, $(1+ε)$-approximate mincut, $(1+ε)$-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).
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