Fully Dynamic $k-\text{Clustering}$ with Fast Update Time and Small Recourse

In the dynamic metric <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{median}$</tex> problem, we wish to maintain a set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> centers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S\subseteq V$</tex> in an input metric space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(V, d)$</tex> that gets updated via point insertions/deletions, so as to minimize the objective <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sum\nolimits_{x\in V}\min\nolimits_{y\in S}d(x, y)$</tex>. The quality of a dynamic algorithm is measured in terms of its approximation ratio, “recourse” (the number of changes in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S$</tex> per update) and “update time” (the time it takes to handle an update). The ultimate goal in this line of research is to obtain a dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(1)$</tex> approximation algorithm with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(1)$</tex> recourse and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k)$</tex> update time. Dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{median}$</tex> is a canonical example of a class of problems known as dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{clustering}$</tex>, that has received significant attention in recent years [Fichtenberger et al, SODA'21], [Bateni et al, SODA'23], [Lacki et al, SODA'24]. To the best of our knowledge, however, all these previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{median}$</tex> in particular, the state-of-the-art results get <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k^{2})$</tex> update time and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(k)$</tex> recourse [Cohen-Addad et al, ICML'19], [Henzinger and Kale, ESA'20], [Bhattacharya et al, NeurIPS'23]. But, this recourse bound of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(k)$</tex> can be trivially obtained by recomputing an optimal solution from scratch after every update, provided we ignore the update time. In addition, the update time of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k^{2})$</tex> is polynomially far away from the desired bound of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k)$</tex>. We come arbitrarily close to resolving the main open question on this topic, with the following results. (I) We develop a new framework of randomized local search that is suitable for adaptation in a dynamic setting. For every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon &gt; 0$</tex>, this gives us a dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{median}$</tex> algorithm with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(k^{\epsilon})$</tex> approximation ratio, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k^{\epsilon})$</tex> recourse and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k^{1+\epsilon})$</tex> update time. This framework also generalizes to dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{clustering}$</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell^{p}$</tex> -norm objectives. As a corollary, we obtain similar bounds for the dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{means}$</tex> problem, and a new trade-off between approximation ratio, recourse and update time for the dynamic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{center}$</tex> problem. (II) If it suffices to maintain only an estimate of the value of the optimal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{median}$</tex> objective, then we obtain a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(1)$</tex> approximation algorithm with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\tilde{O}(k)$</tex> update time. We achieve this result via adapting the Lagrangian Relaxation framework of [Jain and Vazirani, JACM'01], and a facility location algorithm of [Mettu and Plaxton, FOCS'00] in the dynamic setting.