Logarithmic Approximations for Fair k-Set Selection

TL;DR

This paper investigates the fair k-set selection problem, establishing its computational complexity, and proposes approximation algorithms with logarithmic guarantees, applicable to various graph structures and weighted scenarios.

Contribution

It proves NP-hardness for maximum degree 3, provides polynomial solutions for degree 2 and laminar families, and introduces LP-based approximation algorithms with tight bounds.

Findings

NP-hardness for degree 3 bipartite graphs

Polynomial time for degree 2 and laminar families

Logarithmic approximation algorithms with tight bounds

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 $G:=(L\cup R, E)$, 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 $O(\log\Delta)$-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.