Universally-Optimal Distributed Algorithms for Known Topologies

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) Wha...

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Bibliographic Details
Published inarXiv.org
Main Authors Haeupler, Bernhard, Wajc, David, Zuzic, Goran
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 26.12.2023
Subjects
Algorithms
Complexity
Graph theory
Lower bounds
Network topologies
Optimization
Parameters
Shortest-path problems
Topology optimization
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Summary: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.
ISSN:2331-8422