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Optimal Centrality Computations Within Bounded Clique-Width Graphs

Author Guillaume Ducoffe

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Author Details

Guillaume Ducoffe
  • National Institute for Research and Development in Informatics, Bucharest, Romania
  • University of Bucharest, Romania

Funding

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.

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Guillaume Ducoffe. Optimal Centrality Computations Within Bounded Clique-Width Graphs. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.IPEC.2021.16

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 O(2^{O(k)}(n+m)^{1+ε}) time for any ε > 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 O(klog²{n}) bits per vertex and constructible in Õ(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 Õ(kn²)-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Shortest paths
Keywords
  • Clique-width
  • Centralities computation
  • Facility Location problems
  • Distance-labeling scheme
  • Fine-grained complexity in P
  • Graph theory

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