Effect of Gromov-hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

TL;DR

This paper investigates how the hyperbolicity parameter $oldsymbol{}$ influences graph expansion and cut properties, providing constructive bounds and algorithmic implications for $oldsymbol{}$-hyperbolic graphs in network analysis.

Contribution

It introduces the first constructive bounds on node expansions in $oldsymbol{}$-hyperbolic graphs and demonstrates how to efficiently find witnesses and cuts with algorithmic applications.

Findings

Constructive bounds on node expansions as a function of $oldsymbol{}$.

Efficient algorithms to find witnesses for expansion properties.

Methods to identify large families of s-t cuts with few cut-edges.

Abstract

$\delta$-hyperbolic graphs, originally conceived by Gromov in 1987, occur often in many network applications; for fixed $\delta$, such graphs are simply called hyperbolic graphs and include non-trivial interesting classes of "non-expander" graphs. The main motivation of this paper is to investigate the effect of the hyperbolicity measure $\delta$ on expansion and cut-size bounds on graphs (here $\delta$ need not be a constant), and the asymptotic ranges of $\delta$ for which these results may provide improved approximation algorithms for related combinatorial problems. To this effect, we provide constructive bounds on node expansions for $\delta$-hyperbolic graphs as a function of $\delta$, and show that many witnesses (subsets of nodes) for such expansions can be computed efficiently even if the witnesses are required to be nested or sufficiently distinct from each other. To the best of our knowledge, these are the first such constructive bounds proven. We also show how to find a large family of s-t cuts with relatively small number of cut-edges when s and t are sufficiently far apart. We then provide algorithmic consequences of these bounds and their related proof techniques for two problems for $\delta$-hyperbolic graphs (where $\delta$ is a function $f$ of the number of nodes, the exact nature of growth of $f$ being dependent on the particular problem considered).