Matrix Editing Meets Fair Clustering: Parameterized Algorithms and Complexity

TL;DR

The paper investigates the complexity of FDMC, the problem of editing a matrix to obtain at most $r$ distinct rows while ensuring every cluster is color-fair. It proves a strong hardness barrier: even the binary case 2-FDMC is W[1]-hard when parameterized by the fairlet size $ ilde{c}$ plus the budget $k$, underscoring the inadequacy of naive parameterizations. It then delineates tractable regimes via three avenues: imposing additional constraints (e.g., $ ilde{c}>k$ or combining $k$ with $r$), fixed-parameter approximation with a $(5 - 3/ ilde{c})$ factor, and an alternative parameterization using treewidth, including a treewidth-based DP for 2-FDMC. Collectively, these results map a rich complexity landscape and provide practical algorithmic strategies for fair matrix editing under constrained resources. The work also outlines potential generalizations to other fairness notions and higher-domain settings, setting a foundation for further theoretical and applied exploration in fair clustering and matrix editing.

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.

Figures (3)

• Figure 1: (Top) A matrix (left) received 7 edits (center), reducing the number of distinct rows from 6 to 2. Equivalently, the rows are partitioned into clusters (right) with centers 0000 and 1011, respectively. Hamming distances between rows and their respective center equal the number of edits in the middle. (Bottom) Example of matrix editing with fairness colors blue and rose. The center matrix has 2 distinct rows but is not fair. The right matrix is fair, with one cluster consisting of a single fairlet (a blue row and a rose row) and the other cluster consisting of two fairlets.
• Figure 2: Example of an edit graph (right) describing the changes from a matrix $\mathop{\mathrm{\mathbf{M}}}\nolimits$ (left) to a matrix $\mathop{\mathrm{\mathbf{M}}}\nolimits'$ (center). Weights are printed in bold and labels are printed in (brackets). Solid and dashed edges represent changes in the two different fairness colors.
• Figure 3: For the edit graph in \ref{['fig:example_edit_graph']}, the partitions of edges into fair sets of size $\tilde{c} = 2$ are described on the left. (Numbers refer to labels of the edges in \ref{['fig:example_edit_graph']}.) Note that there are no $\mathop{\mathrm{\mathbf{M}}}\nolimits$-fair types, and hence no partitions of the form $P^-_{\tau}$ in this example. The right side depicts the union of two auxiliary graphs for the two color classes, one with the pink solid edges and one with the blue dashed edges.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 5
• Proof 1
• Theorem 5
• Proof 2
• Corollary 1