Accelerated Relax-and-Round for Concave Coverage Problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915990887989248 |
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| author | Fahrbach, Matthew Liaee, Mehraneh Zadimoghaddam, Morteza |
| author_facet | Fahrbach, Matthew Liaee, Mehraneh Zadimoghaddam, Morteza |
| contents | We present an accelerated relax-and-round algorithm for concave coverage problems, which generalize the classic maximum coverage problem. Building on the relax-and-round framework of Barman et al. [STACS 2021], we propose two significant improvements. First, we replace the linear programming (LP) relaxation step with a projected accelerated gradient method applied to a smooth surrogate objective to achieve a $\widetilde{O}(mn \varepsilon^{-1})$ running time. Second, we use a specialized rounding scheme for the hypersimplex that combines the Carathéodory decomposition algorithm in Karalias et al. [NeurIPS 2025] with randomized swap rounding of Chekuri et al. [FOCS 2010]. We prove tight approximation ratios for new reward functions, including a $0.827$-approximation for the logarithmic reward $φ(x) = \log(1 + x)$. Finally, we conduct maximum multi-coverage experiments on synthetic and real-world graphs, demonstrating that our algorithm outperforms approaches that use state-of-the-art LP solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accelerated Relax-and-Round for Concave Coverage Problems Fahrbach, Matthew Liaee, Mehraneh Zadimoghaddam, Morteza Data Structures and Algorithms Machine Learning We present an accelerated relax-and-round algorithm for concave coverage problems, which generalize the classic maximum coverage problem. Building on the relax-and-round framework of Barman et al. [STACS 2021], we propose two significant improvements. First, we replace the linear programming (LP) relaxation step with a projected accelerated gradient method applied to a smooth surrogate objective to achieve a $\widetilde{O}(mn \varepsilon^{-1})$ running time. Second, we use a specialized rounding scheme for the hypersimplex that combines the Carathéodory decomposition algorithm in Karalias et al. [NeurIPS 2025] with randomized swap rounding of Chekuri et al. [FOCS 2010]. We prove tight approximation ratios for new reward functions, including a $0.827$-approximation for the logarithmic reward $φ(x) = \log(1 + x)$. Finally, we conduct maximum multi-coverage experiments on synthetic and real-world graphs, demonstrating that our algorithm outperforms approaches that use state-of-the-art LP solvers. |
| title | Accelerated Relax-and-Round for Concave Coverage Problems |
| topic | Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2605.06900 |