Nathaniel Harms ; Sebastian Wild ; Viktor Zamaraev - Randomized Communication and Implicit Graph Representations

theoretics:11607 - TheoretiCS, October 1, 2025, Volume 4 - https://doi.org/10.46298/theoretics.25.20
Randomized Communication and Implicit Graph RepresentationsArticle

Authors: Nathaniel Harms ORCID; Sebastian Wild ORCID; Viktor Zamaraev ORCID

    We initiate the focused study of constant-cost randomized communication, with emphasis on its connection to graph representations. We observe that constant-cost randomized communication problems are equivalent to hereditary (i.e. closed under taking induced subgraphs) graph classes which admit constant-size adjacency sketches and probabilistic universal graphs (PUGs), which are randomized versions of the well-studied adjacency labeling schemes and induced-universal graphs. This gives a new perspective on long-standing questions about the existence of these objects, including new methods of constructing adjacency labeling schemes.
    We ask three main questions about constant-cost communication, or equivalently, constant-size PUGs: (1) Are there any natural, non-trivial problems aside from Equality and k-Hamming Distance which have constant-cost communication? We provide a number of new examples, including deciding whether two vertices have path-distance at most k in a planar graph, and showing that constant-size PUGs are preserved by the Cartesian product operation. (2) What structures of a problem explain the existence or non-existence of a constant-cost protocol? We show that in many cases a Greater-Than subproblem is such a structure. (3) Is the Equality problem complete for constant-cost randomized communication? We show that it is not: there are constant-cost problems which do not reduce to Equality.

    81 pages. This is the TheoretiCS journal version


    Volume: Volume 4
    Published on: October 1, 2025
    Accepted on: July 18, 2025
    Submitted on: July 19, 2023
    Keywords: Data Structures and Algorithms, Computational Complexity, Discrete Mathematics

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