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Optimal Centrality Computations Within Bounded Clique-Width Graphs

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Abstract

Given an n-vertex m-edge graph G of clique-width at most k, and a corresponding k-expression, we present algorithms for computing some well-known centrality indices (eccentricity and closeness) that run in \({\mathcal {O}}(2^{{\mathcal {O}}(k)}(n+m)^{1+\epsilon })\) time for any \(\epsilon > 0\). Doing so, we can solve various distance problems within the same amount of time, including: the diameter, the center, the Wiener index and the median set. Our run-times match conditional lower bounds of Coudert et al. (SODA’18) under the Strong Exponential-Time Hypothesis. On our way, we get a distance-labeling scheme for n-vertex m-edge graphs of clique-width at most k, using \({\mathcal {O}}(k\log ^2{n})\) bits per vertex and constructible in \(\tilde{\mathcal {O}}(k(n+m))\) time from a given k-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on k in their scheme. As a corollary, we get an \(\tilde{\mathcal {O}}(kn^2)\)-time algorithm for computing All-Pairs Shortest-Paths on n-vertex graphs of clique-width at most k, being given a k-expression. This partially answers an open question of Kratsch and Nelles (STACS’20). Our algorithms work for graphs with non-negative vertex-weights, under two different types of distances studied in the literature. For that, we introduce a new type of orthogonal range query as a side contribution of this work, that might be of independent interest.

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Notes

  1. In all fairness, the labeling scheme of Courcelle and Vanicat can be applied to many more problems than just the computation of the distances in the graph.

  2. This is a slightly different formula than in Lemma 3.3, which is for vertex-weighted graphs.

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  1. National Institute for Research and Development in Informatics, Bucharest, Romania

    Guillaume Ducoffe

  2. Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

    Guillaume Ducoffe

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  1. Guillaume Ducoffe
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This work was supported by project PN-19-37-04-01 “New solutions for complex problems in current ICT research fields based on modelling and optimization”, funded by the Romanian Core Program of the Ministry of Research and Innovation (MCI) 2019-2022. It was also supported by Grant TC ICUB-SSE 15109-26.07.2021, “The complexity landscape of Maximum Matching”. Results of this paper were partially presented at the IPEC’21 conference [27].

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Ducoffe, G. Optimal Centrality Computations Within Bounded Clique-Width Graphs. Algorithmica 84, 3192–3222 (2022). https://doi.org/10.1007/s00453-022-01015-w

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