Brief Announcement: Agreement Tasks in Fault-Prone Synchronous Networks of Arbitrary Structures

Abstract

Consensus is arguably the most studied problem in distributed computing as a whole, and particularly in distributed message-passing settings. Research on consensus has considered various failure types, memory constraints, and much more. Surprisingly, almost all of this work assumes that messages are passed in a complete network, i.e., each process has a direct link to every other process. Set agreement, a relaxed variant of consensus, has also been heavily studied in different settings, yet research on it has also been limited to complete networks. We address this situation by considering consensus and set agreement in general networks, i.e., that can have an arbitrary graph G as their communication graph. We focus on fault-prone networks, where up to t nodes may crash and irrevocably stop communicating, and present upper and lower bounds for such networks. We establish the following collection of results:  
- The consensus algorithm by [Castañeda et al., 2023] is optimal for all graphs, and not only for symmetric graphs. 
- This algorithm can be extended to a generic algorithm for k-set agreement, for every k ≥ 1. For k = 1, our generic algorithm coincides with the existing one for consensus. 
- All these algorithms can be extended to the case where the number t of failures exceeds the connectivity κ of the graph, while the existing consensus algorithm assumed that t < κ.