Wait-free approximate agreement on graphs

Approximate agreement is one of the few variants of consensus that can be solved in a wait-free manner in asynchronous systems where processes communicate by reading and writing to shared memory. In this work, we consider a natural generalisation of approximate agreement on arbitrary undirected connected graphs. Each process is given a vertex of the graph as input and, if non-faulty, must output a vertex such that all the outputs are within distance 1 of one another, and each output value lies on a shortest path between two input values. From prior work, it is known that there is no wait-free algorithm among 𝑛≥3 processes for this problem on any cycle of length 𝑐≥4 , by reduction from 2-set agreement (Castañeda et al. 2018). In this work, we investigate the solvability and complexity of this task on general graphs. We give a new, direct proof of the impossibility of approximate agreement on cycles of length 𝑐≥4 , via a generalisation of Sperner’s Lemma to convex polygons. We also extend the reduction from 2-set agreement to a larger class of graphs, showing that approximate agreement on these graphs is unsolvable. On the positive side, we present a wait-free algorithm for a class of graphs that properly contains the class of chordal graphs.