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Tight Approximations of Degeneracy in Large Graphs

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LATIN 2016: Theoretical Informatics (LATIN 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9644))

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Abstract

Given an n-node m-edge graph G, the degeneracy of graph G and the associated node ordering can be computed in linear time in the RAM model by a greedy algorithm that iteratively removes the node of min-degree [28]. In the semi-streaming model for large graphs, where memory is limited to \(\mathcal {O}(n \,\mathrm{polylog}\,n)\) and edges can only be accessed in sequential passes, the greedy algorithm requires too many passes, so another approach is needed.

In the semi-streaming model, there is a deterministic log-pass algorithm for generating an ordering whose degeneracy approximates the minimum possible to within a factor of \((2+\varepsilon )\) for any constant \(\varepsilon > 0\) [12]. In this paper, we propose a randomized algorithm that improves the approximation factor to \((1+\varepsilon )\) with high probability and needs only a single pass. Our algorithm can be generalized to the model that allows edge deletions, but then it requires more computation and space usage.

The generated node ordering not only yields a \((1+\varepsilon )\)-approximation for the degeneracy but gives constant-factor approximations for arboricity and thickness.

Work supported by CNS-1408782 and IIS-1247750.

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Notes

  1. 1.

    The degeneracy of a graph was originally defined to be the maximum minimum degree among all subgraphs [2, 5–7, 14, 28, 34]. The definition here is a slight modification of the coloring number [5, 6, 14] of a graph, a dual definition of degeneracy. The coloring number of a graph was shown to be one larger than the degeneracy [5, 6, 14], and our definition yields the same value as the original definition of degeneracy.

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  1. Rutgers University, New Brunswick, NJ, 08901, USA

    Martín Farach-Colton & Meng-Tsung Tsai

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  1. Martín Farach-Colton
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  2. Meng-Tsung Tsai
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Correspondence to Meng-Tsung Tsai .

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  1. Carleton University, Ottawa, ON, Ontario, Canada

    Evangelos Kranakis

  2. University Chile, Santiago, Chile

    Gonzalo Navarro

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Farach-Colton, M., Tsai, MT. (2016). Tight Approximations of Degeneracy in Large Graphs. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_32

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  • DOI: https://doi.org/10.1007/978-3-662-49529-2_32

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