Matrix Editing Meets Fair Clustering: Parameterized Algorithms and Complexity

2025-12-03

AI Summary

This paper studies fair mean clustering of discrete vectors, equivalently formulated as editing a colored matrix—via at most $k$ element modifications—into one with few color-balanced rows. We establish tight W[1]-hardness lower bounds, proving that no FPT algorithm exists under classical parameters ($k$, number of colors, number of rows), thereby systematically ruling out fixed-parameter tractability. To overcome this barrier, we propose three novel avenues: (1) designing fixed-parameter approximation algorithms; (2) introducing structural parameters—specifically, treewidth—and developing an FPT algorithm for tree-structured matrices; and (3) establishing a unified modeling framework jointly capturing matrix editing and fair clustering. Our work fully characterizes the computational complexity landscape of the problem, achieving breakthroughs in solvability along three orthogonal dimensions: approximation guarantees, structural constraints, and alternative parameterizations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• Addresses fair clustering of discrete vectors via matrix editing.
• Explores parameterized complexity and algorithmic tractability for fairness.
• Identifies conditions to circumvent computational hardness in fair clustering.

Innovation

• Parameterized algorithms for fair clustering
• Complexity analysis with lower bounds
• Tractability via constraints or approximations