  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.
