Sketching Distances in Monotone Graph Classes

TL;DR

This paper investigates the existence of small, efficient sketches for graph distances and adjacency within monotone graph classes, linking these properties to graph structural parameters like arboricity and expansion.

Contribution

It establishes precise conditions under which constant-size sketches exist for adjacency, exact, and approximate distances in monotone graph classes.

Findings

Constant-size adjacency sketches exist iff the class has bounded arboricity.

Constant-size exact distance sketches exist iff the class has bounded expansion.

Constant-size approximate distance threshold sketches imply bounded expansion.

Abstract

We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized protocol. Equivalently, we consider the problem of assigning constant-size random labels (sketches) to the vertices of a graph, which allow adjacency, exact distance thresholds, or approximate distance thresholds to be computed with high probability from the labels. Our main results are that, for monotone classes of graphs: constant-size adjacency sketches exist if and only if the class has bounded arboricity; constant-size sketches for exact distance thresholds exist if and only if the class has bounded expansion; constant-size approximate distance threshold (ADT) sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has constant-size ADT sketches; and a class may have arbitrarily small expansion without admitting constant-size ADT sketches.