Universally-Optimal Distributed Algorithms for Known Topologies

TL;DR

This paper characterizes the fundamental complexity parameters of distributed optimization on known network topologies and introduces universally optimal algorithms that match these bounds across various problems.

Contribution

It provides tight universal lower bounds for multiple network optimization problems and demonstrates that existing shortcut-based algorithms are universally optimal under known topologies.

Findings

Graph parameters tightly bound problem complexity.

Shortcut-based algorithms are universally optimal with known topologies.

Results apply to MST, min cut, shortest paths, and connectivity.

Abstract

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.