Optimal Centrality Computations Within Bounded Clique-Width Graphs

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 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 O ~ ( k n 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.