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

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Abstract

\(\delta \)-Hyperbolic graphs, originally conceived by Gromov (Essays in group theory. 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).

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Notes

  1. This is in contrast to many research works in this area where one studies the properties of \(\delta \)-hyperbolic graphs assuming \(\delta \) to be fixed.

  2. We will later need to vary the value of \(\alpha \) in our analysis.

  3. Note that if there is no path between nodes p and q in \(G_{-\mathcal {C}}\) then \(\mathrm {dist}_{G_{-\mathcal {C}}}(p,q)=\infty \) and \(\mathcal {B}_{G_{-\mathcal {C}}}\left( p,{\mathrm {dist}_{G_{-\mathcal {C}}}(p,q)}/{2}\right) \) and \(\mathcal {B}_{G_{-\mathcal {C}}}\left( q,{\mathrm {dist}_{G_{-\mathcal {C}}}(p,q)}/{2}\right) \) contain all the nodes reachable from p and q, respectively, in \(G_{-\mathcal {C}}\).

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Acknowledgements

B. DasGupta, N. Mobasheri and F. Yahyanejad thankfully acknowledge supported from NSF Grant IIS-1160995 for this research. M. Karpinski was supported in part by DFG Grants. The problem of investigating expansion properties of \(\delta \)-hyperbolic graphs was raised originally to some of the authors by A. Wigderson.

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Bhaskar Das Gupta, Nasim Mobasheri & Farzane Yahyanejad

  2. Department of Computer Science, University of Bonn, Bonn, 53113, Germany

    Marek Karpinski

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  1. Bhaskar Das Gupta
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Correspondence to Bhaskar Das Gupta.

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Das Gupta, B., Karpinski, M., Mobasheri, N. et al. Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications. Algorithmica 80, 772–800 (2018). https://doi.org/10.1007/s00453-017-0291-7

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