Matrix Editing Meets Fair Clustering: Parameterized Algorithms and Complexity

TL;DR

This paper investigates the computational complexity of fair clustering and matrix editing problems, revealing hardness results and exploring conditions under which these problems become tractable or approximable.

Contribution

It provides a comprehensive complexity analysis of fair clustering and matrix editing, including hardness results and tractability under specific constraints or parameterizations.

Findings

NP-hardness in fair and vanilla settings

No fixed-parameter algorithm for fair clustering in restricted cases

Tractability achieved through additional constraints or alternative parameters

Abstract

We study the computational problem of computing a fair means clustering of discrete vectors, which admits an equivalent formulation as editing a colored matrix into one with few distinct color-balanced rows by changing at most $k$ values. While NP-hard in both the fairness-oblivious and the fair settings, the problem is well-known to admit a fixed-parameter algorithm in the former ``vanilla'' setting. As our first contribution, we exclude an analogous algorithm even for highly restricted fair means clustering instances. We then proceed to obtain a full complexity landscape of the problem, and establish tractability results which capture three means of circumventing our obtained lower bound: placing additional constraints on the problem instances, fixed-parameter approximation, or using an alternative parameterization targeting tree-like matrices.