On Tight FPT Time Approximation Algorithms for k-Clustering Problems

TL;DR

The paper advances the study of fixed-parameter tractable time (FPT) approximation for k-clustering with monotone symmetric norms, providing a unified LP-rounding and sampling framework. It delivers a tight (3+ε)-approximation for minimum-norm capacitated k-clustering and a tight (1+2/(ec)+ε)-approximation for the top-cn norm in uncapacitated settings, parameterized by k and ε, with a simple (3+ε) case for c≤1/e. The results extend to bicriteria guarantees combining k-center and k-median objectives and improve the state of the art for several FPT-time clustering problems, including capacitated k-center in FPT time. The techniques leverage strategic guessing of colors, pivots, radii, and representative sets, together with LP-based rounding that uses occurrence-time vectors to handle top-ell and ordered norms, offering a versatile framework for future k-clustering developments.

Abstract

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.

Figures (1)

• Figure 1: 3 types of connected components of $H$. In the first type, a type-1 color $c$ is connected to $q_c$. Notice that $q_c$ is defined via a representative client $p_c \in R \cap \bar{J}^*_c$. In the second type, we have many type-2 colors $c \in D$ connected to their common $p_c$. $p_c$ for a type-2 color $c$ is defined via a client $j$. In the third type, we have many type-2 colors $c \in D$ ($D'$ in the figure to avoid confusion) connected to their common $p_c$, which is $i^*_{c'}$ for a type-3 color $c'$. Then we have edges $(c', i^*_{c'})$ and $(c', g_c)$ for every $c \in D$ in the component. Solid squares in $T$ are in $S$, and empty squares in $T$ are not in $S$.

Theorems & Definitions (63)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1: Norms
• Definition 2.2: Monotone and Symmetric Norms
• Definition 2.3: Ordered norms
• Lemma 2.4
• Lemma 2.6: Chakrabarty2018
• Definition 2.7: Minimum-Norm $k$-Clustering