Local Search for Fair Max-Min Diversification

2026-09-28

AI Summary

This study addresses the max-min diversification problem subject to partition constraints, aiming to maximize the diversity of selected points while strictly satisfying cardinality limits for each color class. By integrating local search techniques with parameterized algorithm design, the proposed approach handles combinatorial optimization and fairness constraints in metric spaces. This work presents the first constant-factor approximation algorithm for this problem, achieving a running time of f(m)·poly(n) while exactly satisfying all constraints in polynomial time. It overcomes limitations of prior methods that suffered from high approximation ratios, exponential time complexity, or merely expected constraint satisfaction. By unifying parameterized polynomial-time computation with constant approximation guarantees, the framework is further generalized to accommodate arbitrary upper- and lower-bound constraints on subset colors.

Abstract

Given $n$ points in a metric space, Max-Min diversification asks for a subset of $k$ points maximizing the minimum pairwise distance between the selected points. This is arguably the most fundamental notion of diversity with applications across a wide range of domains. We consider this problem under partition constraints, previously studied as Fair Max-Min Diversification (FMMD). Here, each point has a color in $[m]$, and a feasible solution must contain exactly $k_i$ points of color $i$, where $k_1,\ldots,k_m$ are prescribed parameters satisfying $\sum_i k_i=k$. We give the first constant factor approximation for the problem using local search, that runs in time $f(m)\cdot \operatorname{poly}(n)$, in which all constraints are satisfied exactly. All previously known algorithms either provided an $\widetilde Θ(m)$ approximation factor, had running times exponential in the solution size $k$, or satisfied the fairness constraints only approximately or in expectation. We further generalize our result to the problem where each point may belong to an arbitrary subset of colors. Given lower and upper bounds $\ell_i$ and $u_i$ for every color $i$, the goal is to find $k$ points whose color counts satisfy all these bounds while maximizing their diversity.

Problem

• Max-Min Diversification
• Fairness Constraints
• Partition Constraints
• Local Search
• Approximation Algorithm

Innovation

• Fair Max-Min Diversification
• Local Search
• Constant Factor Approximation
• Partition Constraints
• Metric Space