Randomized Communication and Implicit Graph Representations

TL;DR

This paper explores the connection between constant-cost randomized communication protocols and graph representations, introducing new examples and methods for constructing adjacency sketches and universal graphs.

Contribution

It establishes a novel link between randomized communication complexity and hereditary graph classes, providing new constructions and insights into adjacency labeling schemes and universal graphs.

Findings

Identified new problems with constant-cost communication beyond classical examples.

Showed that constant-size probabilistic universal graphs are preserved under Cartesian product.

Demonstrated that the Equality problem is not complete for constant-cost randomized communication.

Abstract

We initiate the focused study of constant-cost randomized communication, with emphasis on its connection to graph representations. We observe that constant-cost randomized communication problems are equivalent to hereditary (i.e. closed under taking induced subgraphs) graph classes which admit constant-size adjacency sketches and probabilistic universal graphs (PUGs), which are randomized versions of the well-studied adjacency labeling schemes and induced-universal graphs. This gives a new perspective on long-standing questions about the existence of these objects, including new methods of constructing adjacency labeling schemes. We ask three main questions about constant-cost communication, or equivalently, constant-size PUGs: (1) Are there any natural, non-trivial problems aside from Equality and k-Hamming Distance which have constant-cost communication? We provide a number of new examples, including deciding whether two vertices have path-distance at most k in a planar graph, and showing that constant-size PUGs are preserved by the Cartesian product operation. (2) What structures of a problem explain the existence or non-existence of a constant-cost protocol? We show that in many cases a Greater-Than subproblem is such a structure. (3) Is the Equality problem complete for constant-cost randomized communication? We show that it is not: there are constant-cost problems which do not reduce to Equality.