The Impossibility of Approximate Agreement on a Larger Class of Graphs

Abstract

Approximate agreement is a variant of consensus in which processes receive input values from a domain and must output values in that domain that are sufficiently close to one another. We study the problem when the input domain is the vertex set of a connected graph. In asynchronous systems where processes communicate using shared registers, there are wait-free approximate agreement algorithms when the graph is a path or a tree, but not when the graph is a cycle of length at least 4. For many graphs, it is unknown whether a wait-free solution for approximate agreement exists. We introduce a set of impossibility conditions and prove that approximate agreement on graphs satisfying these conditions cannot be solved in a wait-free manner. In particular, the graphs of all triangulated d-dimensional spheres that are not cliques, satisfy these conditions. The vertices and edges of an octahedron is an example of such a graph. We also present a family of reductions from approximate agreement on one graph to another graph. This allows us to extend known impossibility results to even more graphs