Logarithmic Approximations for Fair k-Set Selection
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916742164381696 |
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| author | Li, Shi Xu, Chenyang Zhang, Ruilong |
| author_facet | Li, Shi Xu, Chenyang Zhang, Ruilong |
| contents | 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 $Δ$ of the input bipartite graph is $3$, and the problem is in P when $Δ=2$. 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 $O(\logΔ)$-approximation on bipartite graphs with a maximum degree $Δ$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic Approximations for Fair k-Set Selection Li, Shi Xu, Chenyang Zhang, Ruilong Data Structures and Algorithms 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 $Δ$ of the input bipartite graph is $3$, and the problem is in P when $Δ=2$. 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 $O(\logΔ)$-approximation on bipartite graphs with a maximum degree $Δ$. 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. |
| title | Logarithmic Approximations for Fair k-Set Selection |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2505.12123 |