FPT Constant Approximation Algorithms for Colorful Sum of Radii

2025-06-16

AI Summary

This paper studies the Colorful Radius Sum clustering problem: given a point set (P) colored into (omega) classes and outlier budgets (m_i) per class, select (k) centers and assign points to clusters such that at most (m_i) points from class (i) are discarded as outliers, minimizing the sum of cluster radii—the maximum distance from any assigned point in a cluster to its center. This NP-hard problem was introduced by Chekuri et al. (2022), with the prior best approximation ratio being (O(log omega)). We present the first fixed-parameter tractable (FPT) constant-factor approximation algorithm. Our approach first yields a ((2+varepsilon))-approximation via an iterative covering strategy; then, through a reduction to a colorful (k)-center subproblem, we obtain a ((7+varepsilon))-approximation. Crucially, the latter runs in time exponential only in (k)—eliminating exponential dependence on the total outlier budget (m = sum_i m_i)—thereby breaking the previous approximation barrier.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Clustering

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, dots, P_omega$, along with per-class outlier bounds $m_1, m_2, dots, m_omega$, summing to $m$. The goal is to select a subset $mathcal{C} subseteq P$ of $k$ centers and assign points to centers in $mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(log omega)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$.

Problem

• Approximates colorful sum of radii with FPT algorithms
• Minimizes sum of cluster radii with outlier constraints
• Improves runtime by reducing dependency on outliers

Innovation

• FPT constant approximation for colorful radii
• Iterative covering achieves 2+ε approximation
• Colorful k-center improves to 7+ε approximation