Parameterized Approximation for Robust Clustering in Discrete Geometric Spaces

We consider the well-studied Robust \((k, z)\)-Clustering problem, which generalizes the classic \(k\)-Median, \(k\)-Means, and \(k\)-Center problems. Given a constant \(z\ge 1\), the input to Robust \((k, z)\)-Clustering is a set \(P\) of \(n\) weighted points in a metric space \((M,\delta)\) and a positive integer \(k\). Further, each point belongs to one (or more) of the \(m\) many different groups \(S_1,S_2,\ldots,S_m\). Our goal is to find a set \(X\) of \(k\) centers such that \(\max_{i \in [m]} \sum_{p \in S_i} w(p) \delta(p,X)^z\) is minimized. This problem arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness. For polynomial time computation, an approximation factor of \(O(\log m/\log\log m)\) is known [Makarychev, Vakilian, COLT \(2021\)], which is tight under a plausible complexity assumption even in the line metrics. For FPT time, there is a \((3^z+\epsilon)\)-approximation algorithm, which is tight under GAP-ETH [Goyal, Jaiswal, Inf. Proc. Letters, 2023]. Motivated by the tight lower bounds for general discrete metrics, we focus on \emph{geometric} spaces such as the (discrete) high-dimensional Euclidean setting and metrics of low doubling dimension, which play an important role in data analysis applications. First, for a universal constant \(\eta_0 >0.0006\), we devise a \(3^z(1-\eta_{0})\)-factor FPT approximation algorithm for discrete high-dimensional Euclidean spaces thereby bypassing the lower bound for general metrics. We complement this result by showing that even the special case of \(k\)-Center in dimension \(\Theta(\log n)\) is \((\sqrt{3/2}- o(1))\)-hard to approximate for FPT algorithms. Finally, we complete the FPT approximation landscape by designing an FPT \((1+\epsilon)\)-approximation scheme (EPAS) for the metric of sub-logarithmic doubling dimension.