Optimal Centrality Computations Within Bounded Clique-Width Graphs

Guillaume Ducoffe (1, 2)

1 ICI - National Institute for Research and Development in Informatics [Bucharest]
2 UniBuc - University of Bucharest

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 runtimes 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 kexpression. 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.

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]

Guillaume Ducoffe. Optimal Centrality Computations Within Bounded Clique-Width Graphs. Algorithmica, inPress, ⟨10.1007/s00453-022-01015-w⟩. ⟨hal-03746312⟩