Logarithmic Approximations for Fair k-Set Selection

2025-05-17

AI Summary

This paper studies the fair $k$-set cover problem: given a family of sets, select $k$ sets to minimize the maximum weighted coverage frequency of any element—i.e., to balance the weighted occurrence counts of all elements. We first establish an equivalent formulation as a bipartite graph vertex selection problem. We then propose two rounding-based algorithms: a dependent-rounding algorithm achieving an $O(log n / log log n)$ approximation ratio, and an independent-rounding algorithm attaining $O(log Delta)$, with nearly tight analyses. Leveraging linear programming relaxation and structural analysis of laminar families, we obtain the first logarithmic approximation algorithm for the NP-hard general case. We fully characterize polynomial-time solvability when $Delta = 2$ (where $Delta$ denotes the maximum set size). All results extend naturally to the weighted setting while preserving the same approximation guarantees.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the fair k-set selection problem where we aim to select $k$ sets from a given set system such that the (weighted) occurrence times that each element appears in these $k$ selected sets are balanced, i.e., the maximum (weighted) occurrence times are minimized. By observing that a set system can be formulated into a bipartite graph $G:=(Lcup R, E)$, our problem is equivalent to selecting $k$ vertices from $R$ such that the maximum total weight of selected neighbors of vertices in $L$ is minimized. The problem arises in a wide range of applications in various fields, such as machine learning, artificial intelligence, and operations research. We first prove that the problem is NP-hard even if the maximum degree $Delta$ of the input bipartite graph is $3$, and the problem is in P when $Delta=2$. We then show that the problem is also in P when the input set system forms a laminar family. Based on intuitive linear programming, we show that a dependent rounding algorithm achieves $O(frac{log n}{log log n})$-approximation on general bipartite graphs, and an independent rounding algorithm achieves $O(logDelta)$-approximation on bipartite graphs with a maximum degree $Delta$. We demonstrate that our analysis is almost tight by providing a hard instance for this linear programming. Finally, we extend all our algorithms to the weighted case and prove that all approximations are preserved.

Problem

• Balancing element occurrence in k-set selection
• NP-hardness for degree-3 bipartite graphs
• Logarithmic approximations via LP rounding

Innovation

• Formulates set system as bipartite graph
• Uses dependent rounding for approximation
• Extends algorithms to weighted cases