On Tight FPT Time Approximation Algorithms for k-Clustering Problems

Following recent advances in combining approximation algorithms with fixed-parameter tractability (FPT), we study FPT-time approximation algorithms for minimum-norm $k$-clustering problems, parameterized by the number $k$ of open facilities. For the capacitated setting, we give a tight $(3+ε)$-approximation for the general-norm capacitated $k$-clustering problem in FPT-time parameterized by $k$ and $ε$. Prior to our work, such a result was only known for the capacitated $k$-median problem [CL, ICALP, 2019]. As a special case, our result yields an FPT-time $3$-approximation for capacitated $k$-center. The problem has not been studied in the FPT-time setting, with the previous best known polynomial-time approximation ratio being 9 [ABCG, MP, 2015]. In the uncapacitated setting, we consider the $top$-$cn$ norm $k$-clustering problem, where the goal of the problem is to minimize the $top$-$cn$ norm of the connection distance vector. Our main result is a tight $\big(1 + \frac 2{ec} + ε\big)$-approximation algorithm for the problem with $c \in \big(\frac1e, 1\big]$. (For the case $c \leq \frac1e$, there is a simple tight $(3+ε)$-approximation.) Our framework can be easily extended to give a tight $\left(3, 1+\frac2e + ε\right)$-bicriteria approximation for the ($k$-center, $k$-median) problem in FPT time, improving the previous best polynomial-time $(4, 8)$ guarantee [AB, WAOA, 2017]. All results are based on a unified framework: computing a $(1+ε)$-approximate solution using $O\left(\frac{k\log n}ε\right)$ facilities $S$ via LP rounding, sampling a few client representatives $R$ based on the solution $S$, guessing a few pivots from $S \cup R$ and some radius information on the pivots, and solving the problem using the guesses. We believe this framework can lead to further results on $k$-clustering problems.