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

Universally-Optimal Distributed Algorithms for Known Topologies