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Modification-Fair Cluster Editing

Vincent Froese, Leon Kellerhals, Rolf Niedermeier

TL;DR

This work introduces modification fairness for Cluster Editing on two-color graphs, enforcing a process-oriented fairness constraint that balances the average number of edits across color groups. It formalizes Delta_ed(S) and analyzes the problem’s complexity, proving NP-hardness with ETH-based lower bounds while providing an FPT algorithm in the total number of edits k and a randomized approach for small numbers of mono-colored edits. The authors also prove NP-hardness for a related cluster transformation problem and corroborate their theoretical results with an empirical study on real networks, showing that the price of fairness is typically modest unless perfect fairness is required. Overall, the paper establishes both theoretical foundations and practical insights for fairness-aware graph clustering, and points to promising directions for extending to more colors and combining process- with output-oriented fairness notions.

Abstract

The classic Cluster Editing problem (also known as Correlation Clustering) asks to transform a given graph into a disjoint union of cliques (clusters) by a small number of edge modifications. When applied to vertex-colored graphs (the colors representing subgroups), standard algorithms for the NP-hard Cluster Editing problem may yield solutions that are biased towards subgroups of data (e.g., demographic groups), measured in the number of modifications incident to the members of the subgroups. We propose a modification fairness constraint which ensures that the number of edits incident to each subgroup is proportional to its size. To start with, we study Modification-Fair Cluster Editing for graphs with two vertex colors. We show that the problem is NP-hard even if one may only insert edges within a subgroup; note that in the classic "non-fair" setting, this case is trivially polynomial-time solvable. However, in the more general editing form, the modification-fair variant remains fixed-parameter tractable with respect to the number of edge edits. We complement these and further theoretical results with an empirical analysis of our model on real-world social networks where we find that the price of modification-fairness is surprisingly low, that is, the cost of optimal modification-fair solutions differs from the cost of optimal "non-fair" solutions only by a small percentage.

Modification-Fair Cluster Editing

TL;DR

This work introduces modification fairness for Cluster Editing on two-color graphs, enforcing a process-oriented fairness constraint that balances the average number of edits across color groups. It formalizes Delta_ed(S) and analyzes the problem’s complexity, proving NP-hardness with ETH-based lower bounds while providing an FPT algorithm in the total number of edits k and a randomized approach for small numbers of mono-colored edits. The authors also prove NP-hardness for a related cluster transformation problem and corroborate their theoretical results with an empirical study on real networks, showing that the price of fairness is typically modest unless perfect fairness is required. Overall, the paper establishes both theoretical foundations and practical insights for fairness-aware graph clustering, and points to promising directions for extending to more colors and combining process- with output-oriented fairness notions.

Abstract

The classic Cluster Editing problem (also known as Correlation Clustering) asks to transform a given graph into a disjoint union of cliques (clusters) by a small number of edge modifications. When applied to vertex-colored graphs (the colors representing subgroups), standard algorithms for the NP-hard Cluster Editing problem may yield solutions that are biased towards subgroups of data (e.g., demographic groups), measured in the number of modifications incident to the members of the subgroups. We propose a modification fairness constraint which ensures that the number of edits incident to each subgroup is proportional to its size. To start with, we study Modification-Fair Cluster Editing for graphs with two vertex colors. We show that the problem is NP-hard even if one may only insert edges within a subgroup; note that in the classic "non-fair" setting, this case is trivially polynomial-time solvable. However, in the more general editing form, the modification-fair variant remains fixed-parameter tractable with respect to the number of edge edits. We complement these and further theoretical results with an empirical analysis of our model on real-world social networks where we find that the price of modification-fairness is surprisingly low, that is, the cost of optimal modification-fair solutions differs from the cost of optimal "non-fair" solutions only by a small percentage.
Paper Structure (13 sections, 18 theorems, 20 equations, 4 figures, 3 tables)

This paper contains 13 sections, 18 theorems, 20 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work.
  3. Our contributions.
  4. Problem definition and initial observations.
  5. Modification Fairness: Complexity
  6. Transforming Cluster Graphs
  7. Empirical Insights into the Price of Fairness
  8. Setup.
  9. Datasets.
  10. How fair is the colorblind variant?
  11. The price of fairness
  12. Discussion.
  13. Conclusion

Key Result

Theorem 2

Mod-i-fi-ca-tion-Fair Cluster Editing and Mod-i-fi-ca-ti-on-Fair Cluster Deletion are $\mathrm{NP}$-hard for arbitrary $\delta\ge 0$ and solvable neither in $2^{o(k)}\cdot |V(G)|^{\mathcal{O}(1)}$ nor in $2^{o(\abs{V(G)} + \abs{E(G)})}$ time unless the ETH fails. This also holds

Figures (4)

  • Figure 1: An exemplary graph $G$ with blue (dark) and red (light) vertices (a) and two transformations of $G$ into a cluster graph (b), (c). Inserted edges are marked green (thick), deleted edges are green and dashed. (b) A transformation of minimum size with five modifications. Eight modifications are incident to blue, two modifications are incident to red, so the average number of modifications to blue (red) vertices is $8/4$ ($2/5$), and the difference is $8/5$. (c) Another minimum-size transformation in which the modifications are more balanced between blue and red (difference $7/10$).
  • Figure 2: How fair is the "non-fair" variant? We compare the normed ($\Delta_{\mathrm{norm}}$) and average ($\Delta_{\mathrm{avg}}$) modification fairness of the optimal solution for standard Cluster Editing to the number $n$ of vertices and the ratio $p$ red vertices (color). Facebook instances are displayed as triangles, Amazon instances are displayed as circles. As the Facebook instances admit four distinct values of $n$, we add a random jiggle to make similar entries more distinguishable, that is, for Facebook instances, we display $n+r$ for $r \in [-5,5]$ chosen uniformly at random. The average modification fairness (right) is displayed on a log scale; we add $10^{-4}$ to each entry to make entries with $\Delta_{\mathrm{avg}} = 0$ visible.
  • Figure 3: How many extra edits do we need to be fair? Each heat map cell contains the mean percentage by which a minimum solution with fairness of $\delta_{\mathrm{norm}}$ is larger than the colorblind solution for Sets 1--4 of the Amazon (A) and Facebook (F) graphs, see \ref{['tab:dataset']}.
  • Figure 4: We compare for each instance the size of a minimum colorblind solution ($x$-axis) with the percentage by which the minimum fair solution is larger ($y$-axis) for the set of Amazon and Facebook graphs. Instances whose percentage was above $100$ (roughly $1.2\%$ of the Amazon and $2.5\%$ of the Facebook instances) are not displayed. The instances are colored by the ILP gap --- note that the coloring follows a logarithmic scale (we added $10^{-3}$ to each gap so as to color zero gaps as well).

Theorems & Definitions (34)

  • proof
  • Theorem 2
  • Proposition 3: KU12
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 7
  • Lemma 8
  • proof
  • ...and 24 more