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Individual Fairness in Graph Decomposition

Kamesh Munagala, Govind S. Sankar

TL;DR

This work studies randomized low-diameter decompositions (LDDs) of planar graphs with the goal of individual fairness, ensuring comparable separation probabilities for pairs at similar distances measured by $\rho_{uv}=d(u,v)/R$. It analyzes the classical KPR method and introduces variants that balance connectivity (CON), compression (COMP), and fairness, including an edge-fair procedure with $\Pr[u,v\text{ separated}]=\Theta(1/R)$ for edges and a diameter bound $O(R)$, as well as general-distance techniques yielding $f(\rho)=O(\rho)$ and positive $g(\rho)$ depending on parameters. A key theoretical contribution links attainable fairness to metric embeddings by showing that achieving $g(\rho)=\Omega(\rho)$ would imply a constant-distortion embedding of planar graphs into $\ell_1$, a major open question in the field, while practical algorithms offer meaningful fairness with trade-offs when COMP or CON are relaxed. The empirical results on precinct graphs demonstrate improved edge fairness and cluster counts consistent with the theory, suggesting the approach can be a useful pre-processing step for fair redistricting and community-preserving clustering in real networks.

Abstract

In this paper, we consider classic randomized low diameter decomposition procedures for planar graphs that obtain connected clusters which are cohesive in that close-by pairs of nodes are assigned to the same cluster with high probability. We require the additional aspect of individual fairness - pairs of nodes at comparable distances should be separated with comparable probability. We show that classic decomposition procedures do not satisfy this property. We present novel algorithms that achieve various trade-offs between this property and additional desiderata of connectivity of the clusters and optimality in the number of clusters. We show that our individual fairness bounds may be difficult to improve by tying the improvement to resolving a major open question in metric embeddings. We finally show the efficacy of our algorithms on real planar networks modeling congressional redistricting.

Individual Fairness in Graph Decomposition

TL;DR

This work studies randomized low-diameter decompositions (LDDs) of planar graphs with the goal of individual fairness, ensuring comparable separation probabilities for pairs at similar distances measured by . It analyzes the classical KPR method and introduces variants that balance connectivity (CON), compression (COMP), and fairness, including an edge-fair procedure with for edges and a diameter bound , as well as general-distance techniques yielding and positive depending on parameters. A key theoretical contribution links attainable fairness to metric embeddings by showing that achieving would imply a constant-distortion embedding of planar graphs into , a major open question in the field, while practical algorithms offer meaningful fairness with trade-offs when COMP or CON are relaxed. The empirical results on precinct graphs demonstrate improved edge fairness and cluster counts consistent with the theory, suggesting the approach can be a useful pre-processing step for fair redistricting and community-preserving clustering in real networks.

Abstract

In this paper, we consider classic randomized low diameter decomposition procedures for planar graphs that obtain connected clusters which are cohesive in that close-by pairs of nodes are assigned to the same cluster with high probability. We require the additional aspect of individual fairness - pairs of nodes at comparable distances should be separated with comparable probability. We show that classic decomposition procedures do not satisfy this property. We present novel algorithms that achieve various trade-offs between this property and additional desiderata of connectivity of the clusters and optimality in the number of clusters. We show that our individual fairness bounds may be difficult to improve by tying the improvement to resolving a major open question in metric embeddings. We finally show the efficacy of our algorithms on real planar networks modeling congressional redistricting.
Paper Structure (39 sections, 20 theorems, 22 equations, 9 figures, 2 tables, 5 algorithms)

This paper contains 39 sections, 20 theorems, 22 equations, 9 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Motivation for Individual Fairness
  3. Randomized LDD in Planar Graphs
  4. Individual Fairness in LDD
  5. Existing Methods can Fail.
  6. Achieving Individual Fairness: Our Results
  7. Fairness for Edges.
  8. Lower Bound on Individual Fairness.
  9. Positive Results for General Distances.
  10. Empirical Results.
  11. Related Work
  12. Low Diameter Decomposition.
  13. Fairness.
  14. Clustering.
  15. The KPR Partitioning Algorithm
  16. ...and 24 more sections

Key Result

Theorem 1.2

For any $n\in \mathbb{N}$ and any distance $1\leq d\leq \frac{\sqrt{n}}{3}$, there exists a planar graph $G_d$ and vertices $u,v\in V(G)$ with $d(u,v)=d$ such that alg:kpr with parameter $R > 2d$ never separates $u$ and $v$.

Figures (9)

  • Figure 1: An example of the event $E_1\cap L_1$ in $T_1$ (red). Both the cuts made by $k_1,k_2$ are within $2\hat{d}$ levels of both $u,v$, which are themselves within $\hat{d}$ levels of each other.
  • Figure 2: Histograms of edge separation probabilities across 3000 runs of \ref{['alg:kpr-randwts']} on NC, MD, and PA.
  • Figure 3: Box plot of the number of clusters across 200 runs of \ref{['alg:kpr-randwts']} on NC, MD, and PA. The normalized number of clusters refers to the number of clusters as a fraction of the baseline $\frac{n}{R}$.
  • Figure 4: Box plot of the maximum diameters of clusters for the states of NC, MD, and PA across 200 clusterings. The diameters are shown as a fraction of $R$.
  • Figure 5: Counterexample for $d=1$.
  • ...and 4 more figures

Theorems & Definitions (49)

  • Definition 1.1: Low Diameter Decomposition
  • Example 1
  • Theorem 1.2: Proved in \ref{['sec:kpr-negative']}
  • Theorem 1.3: Proved in \ref{['sec:alg-randwts']}
  • Theorem 1.4: Proved in \ref{['sec:equiv']}
  • Theorem 1.5: Proved in \ref{['sec:generallb1']}
  • Theorem 1.6: Proved in \ref{['sec:rand-radius']}
  • Theorem 1.7: Proved in \ref{['sec:equiv0']}
  • Theorem 2.1: KPRgoemans-lecturenotes
  • Lemma 2.2: (COMP)
  • ...and 39 more