Faster Randomized and Deterministic k-Clustering on Graphs

In this paper, we study the $(k,z)$-clustering and $k$-center problems on graphs, where $(k,z)$-clustering generalizes the $k$-median ($z=1$) and $k$-means ($z=2$) problems. We obtain the following main results. Our first contribution is the first deterministic algorithm for $k$-center on graphs that achieves a $(2+\varepsilon)$-approximation in $\tilde{O}(m)$ time. This affirmatively resolves an open problem raised by Abboud, Cohen-Addad, Lee, and Manurangsi [SOSA 2023]. Our techniques also extend to the $k$-center with outliers problem, where up to $t$ points may be discarded. Our second contribution is a randomized algorithm for $(k,z)$-clustering on graphs that achieves an $O(1)$-approximation in $\tilde{O}(m)$ time, which in particular covers $k$-median ($z=1$) and $k$-means ($z=2$). Prior to this work, an $\tilde{O}(m)$-time randomized algorithm was known for $k$-median by Thorup [SIAM J. Comput. 2005], and a recent work of Jiang, Jin, Lou, and Lu [2026] achieves $m^{1+o(1)}$ time for general $z$ via local search. Finally, we design a deterministic algorithm for $(k,z)$-clustering on graphs that achieves an $O(\mathrm{poly}(c))$-approximation in $\tilde{O}(m^{1+1/c})$ time, for a positive parameter $c$. To obtain this result, we use techniques from the Thorup-Zwick distance oracle [JACM 2005]; this technical connection may be of independent interest, considering the wide application of distance oracles in various computational settings. Most of our algorithms are incremental, in the sense that for any given parameter $k$, they return a sequence of centers such that every prefix of length $\ell \leq k$ yields a constant-factor approximate solution to the $\ell$-clustering problem.