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Solvability of Approximate Agreement on Graphs and Simplicial Complexes

Authors Joel Rybicki , Yaroslav Verbitsky

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LIPIcs.DISC.2026.7.pdf
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ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • approximate agreement
  • impossibility results
  • algebraic topology

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Abstract

Approximate agreement tasks on graphs are discrete relaxations of consensus, where each process in a distributed system is given as input a vertex on a graph G, and processes have to output vertices that lie on a clique of G contained in the convex hull of the input vertices. Although such tasks have been widely studied in a variety of models, graph classes and notions of convexity, it remains largely open for which classes of graphs these problems are solvable in asynchronous systems.
In this work, we give a complete topological characterisation of the t-resilient solvability of approximate agreement on graphs and simplicial complexes in asynchronous shared-memory systems with read-write registers. As a result, we answer several open problems related to different variants of approximate agreement on graphs. For example, we give the first proof of Ledent’s conjecture [PODC 2021] about the wait-free solvability of clique agreement.
In fact, we show a more general result: clique agreement is t-resilient solvable on a graph G if and only if its clique complex is (t-1)-connected in the homotopical sense. We also show that clique and monophonic agreement are solvable on the same class of graphs, but there exists a separation between monophonic and geodesic agreement, answering a question by Alistarh et al. [TCS 2023]. In the message-passing setting, our results imply new resilience bounds for asynchronous approximate agreement and round lower bounds for synchronous approximate agreement on graphs.

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Joel Rybicki and Yaroslav Verbitsky. Solvability of Approximate Agreement on Graphs and Simplicial Complexes. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.7

Author Details

Joel Rybicki
  • Humboldt-Universität zu Berlin, Germany
Yaroslav Verbitsky
  • Humboldt-Universität zu Berlin, Germany

Acknowledgements

Parts of this work are based on the Master’s thesis of the second author [Yaroslav Verbitsky, 2026]. We are grateful to Faith Ellen, Florian Kaufmann, Laurence Kluge, Jakob Sauer and Chris Wendl for many helpful discussions. We also thank the anonymous reviewers for their comments.

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