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Communication Dans Un Congrès Année : 2021

Optimal centrality computations within bounded clique-width graphs

Résumé

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(k log^2 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(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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hal-03445484 , version 1 (24-11-2021)

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Guillaume Ducoffe. Optimal centrality computations within bounded clique-width graphs. 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), Sep 2021, Lisbon (virtual event), Portugal. ⟨10.4230/LIPIcs.IPEC.2021.16⟩. ⟨hal-03445484⟩
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