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