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Parameterized Aspects of Triangle Enumeration

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Fundamentals of Computation Theory (FCT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10472))

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Abstract

Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, “subcubic” parameterized solving algorithms) as well as negative results (uselessness in terms of possibility of “faster” parameterized algorithms of certain parameters such as diameter).

A full version is available at https://arxiv.org/abs/1702.06548.

T. Fluschnik—Supported by the DFG, project DAMM (NI 369/13-2).

A. Nichterlein—Supported by a postdoc fellowship of the DAAD while at Durham University, UK.

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Notes

  1. 1.

    \(\omega \) is a placeholder for the best known \(n\times n\)-matrix multiplication exponent.

  2. 2.

    Degeneracy measures graph sparseness. A graph G has degeneracy d if every subgraph contains a vertex of degree at most d; thus G contains at most \(n\cdot d\) edges.

  3. 3.

    The 3SUM problem asks whether a given set S of n integers contains three integers \(a, b, c \in S\) summing up to 0. The 3SUM-conjecture states that for any constant \(\varepsilon > 0\) there is no \(O(n^{2-\varepsilon })\)-time algorithm solving 3SUM. The connection between 3SUM and listing/detecting triangles is well studied [24, 29].

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    Matthias Bentert, Till Fluschnik, André Nichterlein & Rolf Niedermeier

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  1. Matthias Bentert
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Correspondence to Matthias Bentert .

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Editors and Affiliations

  1. CNRS, University of Bordeaux, Talence cedex, France

    Ralf Klasing

  2. LaBRI, University of Bordeaux, Talence cedex, France

    Marc Zeitoun

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Bentert, M., Fluschnik, T., Nichterlein, A., Niedermeier, R. (2017). Parameterized Aspects of Triangle Enumeration. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_9

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  • DOI: https://doi.org/10.1007/978-3-662-55751-8_9

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