Byzantine Approximate Agreement on Graphs

Authors Thomas Nowak , Joel Rybicki

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Thomas Nowak
  • Université Paris-Sud, France
  • Centre National de la Recherche Scientifique, Paris, France
Joel Rybicki
  • Institute of Science and Technology Austria, Klosterneuburg, Austria


We thank the anonymous reviewers for their helpful comments and Janne H. Korhonen for many discussions on this work. We also wish to thank the participants of the Helsinki Workshop on Theory of Distributed Computing 2018 and the Metastability workshop in Mainz 2018 for discussions that lead to the problem of approximate agreement on graphs.

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Thomas Nowak and Joel Rybicki. Byzantine Approximate Agreement on Graphs. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors' input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • consensus
  • approximate agreement
  • Byzantine faults
  • chordal graphs
  • lattice agreement


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