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.
In a typical matrix modification problem, we are given a matrix M and are tasked with modifying it into some matrix M ′ satisfying a specified desirable property. Matrix modification problems arise in a broad range of research contexts directly related to artificial intelligence and machine learning, prominently including recommender systems and data recovery (Candès and Plan 2010;Candès and Recht 2012;Elhamifar and Vidal 2013) but also occurring in, e.g., Markov inference (Roth and Yih 2005) and computational social choice (Bredereck et al. 2014). Matrix completion is perhaps the most classical example of matrix modification: there, certain entries in the provided matrix M are marked as "missing" and the task is to complete the missing entries. A second classical example-one which will be the focus of our interests here-is matrix editing, where we are allowed to alter at most k entries of a (complete) matrix M in order to achieve the sought-after property.
The vast majority of matrix completion and matrix editing problems are known to be NP-hard, leading to the investigation of these problems using the more refined parameterized complexity paradigm (Cygan et al. 2015). There, the general aim is to circumvent the intractability of problems by designing algorithms with running times which are not exponential in the whole input size, but only exponential in some well-defined integer parameters of the input. From a complexity-theoretic perspective, we ask for which natural parameters p one can obtain an algorithm solving the problem in time f (p) • n O(1) , where f is a computable function and n the input size; such algorithms are called fixed-parameter and form a weaker (but still desirable) baseline of tractability than polynomial-time algorithms.
The parameterized complexity of matrix completion was first investigated by Ganian, Kanj, Ordyniak and Szeider (Ganian et al. 2018), who targeted the two fundamental cases where M ′ must adhere to an input-specified bound r on either the rank, or the number of distinct rows. Subsequent works in the completion setting then considered a variety of different constraints on the output matrix M ′ (Eiben et al. 2021(Eiben et al. , 2023a;;Ganian et al. 2022;Koana, Froese, andNiedermeier 2020, 2023). For matrix editing, Fomin, Golovach and Panolan (Fomin, Golovach, and Panolan 2020) studied the parameterized complexity of the two problems analogous to those considered in the completion setting (Ganian et al. 2018) and as their main positive result obtained a fixedparameter algorithm for the task of editing a binary matrix to achieve at most r distinct rows, parameterized by the budget k on the number of altered entries (i.e., edits). 1 This task is particularly interesting, as it precisely corresponds to the classical BINARY MEANS CLUSTERING problem (Kleinberg, Papadimitriou, and Raghavan 2004;Ostrovsky and Rabani 2002)-a discrete counterpart to the means clustering that is frequently used on real-valued data in machine learning (Charikar et al. 2023;Marom and Feldman 2019;Zhang, Lange, and Xu 2020). Intuitively, the reason the editing and clustering tasks coincide is that each time we edit a row ⃗ v to its final value ⃗ w, the number of edits is equal to the cost of placing a data point ⃗ v into a cluster centered at ⃗ w; see Figure 1 (Top) for an illustration.
In this article, we investigate the computational complexity of the same task of editing a matrix to achieve at most r distinct rows, but in the presence of a fairness constraint. The reason for considering fairness in this setting is directly tied to the clustering perspective, where requiring each of the clusters to be “fair” is equivalent to ensuring that the (at most r) distinct rows in M ′ are “fair”. Fair clustering has become an increasingly important research topic in recent years (Amagata 2024;Backurs et al. 2019;Bandyapadhyay, Fomin, and Simonov 2024;Dickerson et al. 2023), starting from the pioneering paper of Chierichetti, Kumar, Lattanzi and Vassilvitskii (Chierichetti et al. 2017). The fairness constraint we adopt here is the same as in the latter foundational work: each row is equipped with a specified color (representing an aspect of that data point that should be proportionately represented in clusters) and each cluster must admit a partitioning into fairlets, which are minimum sets of colored elements exhibiting the same color ratio as M. (For example, for two colors with 1:1 ratio, a fairlet contains one element of each color-see Fig. 1 (Bottom).) We remark that while this “canonical” fairness constraint has been used in several related works (Ahmadian et al. 2020;Bandyapadhyay, Fomin, and Simonov 2024;Casel et al. 2023), we also discuss possible extensions of our results to other fairness notions in Section 7.
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