Solvability of Approximate Agreement on Graphs and Simplicial Complexes

2026-06-23

AI Summary

This work investigates the solvability of approximate agreement tasks over graph classes in asynchronous distributed systems. By leveraging connectivity theory from algebraic topology and unifying shared-memory and message-passing models, it establishes—for the first time—the necessary and sufficient condition for t-resilient solvability of clique agreement on a graph G: the clique complex of G must be (t−1)-connected. This result fully characterizes the solvability of clique agreement over graphs, confirms Ledent’s conjecture, and unifies the classes of graphs on which clique agreement and monotonic path agreement are solvable. Moreover, it reveals fundamental differences in solvability under distinct notions of convexity—such as monotonic versus geodesic—and yields novel resilience bounds for asynchronous systems as well as round complexity lower bounds for synchronous settings.

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.

Problem

• approximate agreement
• graphs
• asynchronous systems
• clique agreement
• solvable

Innovation

• approximate agreement
• topological characterization
• clique complex
• t-resilient solvability
• asynchronous distributed systems