Low Cost, Fair, and Representative Committees in a Metric Space

TL;DR

The paper addresses selecting a size-$k$ representative committee from $n$ agents in a metric space to minimize the sum of distances to all agents, a objective that can clash with fairness. It develops an algorithmic framework that blends greedy capture and expanding approvals ideas to achieve fairness notions PRF and NORP, and introduces NORP as a new anti-overrepresentation axiom. The authors prove that PRF, NORP, and metric Justified Representation can be satisfied simultaneously with a constant-factor bound to the optimum (4-approx), and they also provide a spectrum of tradeoffs allowing 2-approximate fairness variants with different fairness guarantees. They further show that certain fairness constraints are inherently incompatible with improving the cost beyond a 2-approximation, and they offer approximations and reductions for broader settings, including when $k$ does not divide $n$. Together, these results demonstrate that low-cost, fair, and representative committees are achievable in general metric spaces under the proposed axioms, with practical algorithms and clear tradeoffs.

Abstract

We study the problem of selecting a representative committee of $k$ agents from a collection of $n$ agents in a common metric space. This problem is related to choosing $k$ facilities in facility location and $k$-median problems. However, unlike in more traditional facility location where each agent only cares about the closest selected facility, in the settings we consider each agent desires that all selected committee members are close to them. More precisely, we look at the sum objective, in which the goal is to minimize the total distance from all agents to all members of the chosen committee. We show that it is always possible to find a committee which is both low-cost according to this objective, and also fair according to many existing notions of fairness and proportionality defined for clustering settings. Moreover, we introduce a new desirable axiom for representative committees we call NORP, which prevents over-representation of any subset of agents. While all existing algorithms for fair committee selection do not satisfy this intuitive property, we provide new algorithms which form simultaneously low-cost, fair, and NORP solutions, thus showing that it is always possible to form low-cost, fair, and representative committees for our settings.

Figures (6)

• Figure 1: An example with a large number of agents in the plane and $k=3$. We have three large groups of $\frac{n}{k}-1$ agents placed symmetrically around the origin, with $3$ agents at the origin. Then standard fairness axioms would allow all three representatives to be placed in the center.
• Figure 2: Example illustrating the incompatibility of PRF, NORP, and $(2-\epsilon)$-approximate solutions.
• Figure 3: Example where Algorithm \ref{['alg:alg2']} gives a solution that is not NORP. $k=6$ and there are $6$ central agents that are all a distance $\epsilon$ from each other; each central agent has a spoke with $\frac{n}{k}-1$ agents a distance $1$ away, all of which are also distance $\epsilon$ from each other. Previous fairness axioms would allow representatives to be placed in the inner clique. This is unfair since the $k$ agents in the middle have a large number of representatives within a distance of $\epsilon$, yet only $6$ agents. A fairer solution (that satisfies all usual fairness properties) would be to place the representatives on the outermost $5$ points with $\frac{n}{k}-1$ agents and one representative in the middle.
• Figure 4: Example where Algorithm \ref{['alg:alg4']} does not satisfy PRF. There are 102 agents in total, broken into three groups. Three agents in the center on top of each other, 65 on the left and 34 on the right. We use $K_r$ to denote a complete graph with $r$ nodes. The 65 on the left are $2-2\epsilon$ from each other and the 34 on the right are $\epsilon$ from each other. All agents in $K_{65}$ and $K_{34}$ are $2-2\epsilon$ from each other. The three agents in the middle are $1$ and $1+\epsilon$ away from the two groups. The three middle agents are the optimal solution for $k=3$ and $\frac{n}{k} = 34$. The algorithm instead picks one central representative and two on the left. Then the group on the right does not have a representative within distance $\epsilon$, which is the diameter of that set.
• Figure 5: In the above example we have $n-2$ agents on the left hand side each distance $0$ from each other and a single agent each in the middle and on the right. Standard Truncated Greedy Capture (along with Greedy Capture and Expanding Approvals if we place extra representatives on the left) would first place $k-1$ representatives on the left, then the last representative in the middle. The optimal solution, however, places all representatives on the left. The cost of the solution returned by the greedy capture algorithm is at least $(k-1)(2-\epsilon) + (n-2)(1-\epsilon)$. The cost of the optimal solution is $k(2-\epsilon)$. As $n \to \infty$ the approximation factor becomes arbitrarily large.

Theorems & Definitions (66)

• Definition 1.1
• Definition 1.2
• Definition 3.1
• Proposition 4.1
• proof
• Lemma 4.2
• proof
• Proposition 4.3
• proof
• Corollary 4.4