Many distributed optimization algorithms achieve existentially-optimal running times, meaning that there exists some pathological worst-case topology on which no algorithm can do better. Still, most networks of interest allow for exponentially faster algorithms. This motivates two questions: (1) What network topology parameters determine the complexity of distributed optimization? (2) Are there universally-optimal algorithms that are as fast as possible on every topology? We resolve these 25-year-old open problems in the known-topology setting (i.e., supported CONGEST) for a wide class of global network optimization problems including MST, $(1+\varepsilon)$-min cut, various approximate shortest paths problems, sub-graph connectivity, etc. In particular, we provide several (equivalent) graph parameters and show they are tight universal lower bounds for the above problems, fully characterizing their inherent complexity. Our results also imply that algorithms based on the low-congestion shortcut framework match the above lower bound, making them universally optimal if shortcuts are efficiently approximable. We leverage a recent result in hop-constrained oblivious routing to show this is the case if the topology is known -- giving universally-optimal algorithms for all above problems.

翻译：许多分布式优化算法实现了存在性最优运行时间，即存在某些病态最差拓扑使得任何算法都无法超越其性能。然而，大多数实际网络允许指数级更快的算法。这引出了两个问题：（1）哪些网络拓扑参数决定了分布式优化的复杂度？（2）是否存在在任意拓扑上都能达到最快速度的通用最优算法？我们在已知拓扑设置（即支持CONGEST模型）下，针对包括最小生成树（MST）、$(1+\varepsilon)$-最小割、各类近似最短路径问题、子图连通性等在内的广泛全局网络优化问题，解决了这些已存在25年的开放性问题。具体而言，我们提出了若干（等价的）图参数，并证明它们构成了上述问题的严格通用下界，完全刻画了其内在复杂度。我们的结果还表明，基于低拥塞捷径框架的算法能达到上述下界，从而在捷径可高效近似时成为通用最优算法。我们利用近期关于跳数约束的 oblivious 路由的研究成果，证明了在拓扑已知的情况下这一条件成立——从而为上述所有问题提供了通用最优算法。