Adding a referee to an interconnection network: What can(not) be computed in one round

TL;DR

This paper investigates the computational power of a restricted distributed model where nodes send limited information to a central referee, showing it cannot decide certain properties in general but can fully determine specific graph classes.

Contribution

It introduces a new model close to communication complexity, analyzing what graph properties can be computed with limited local information sent to a referee.

Findings

Cannot decide if G contains a triangle or square in general

Referee can fully reconstruct G for certain graph classes

Open questions remain on connectivity for arbitrary graphs

Abstract

In this paper we ask which properties of a distributed network can be computed from a little amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical CONGEST (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lokam. More precisely, each of these n nodes -which only knows its own ID and the IDs of its neighbors- is allowed to send a message of O(log n) bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology G? We show that simple questions like, "does G contain a square?", "does G contain a triangle?" or "Is the diameter of G at most 3? cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of G when G belongs to many graph classes such as planar graphs, bounded treewidth graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity of arbitrary graphs.