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 $\varphi(x) = \log(1 + x)$. 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.

翻译：我们提出了一种针对凹覆盖问题的加速松弛舍入算法，该问题推广了经典的最大覆盖问题。基于Barman等人[STACS 2021]的松弛舍入框架，我们提出了两项重要改进。首先，我们采用投影加速梯度方法处理光滑替代目标函数，替代线性规划（LP）松弛步骤，实现了$\widetilde{O}(mn \varepsilon^{-1})$的运行时间复杂度。其次，我们针对超单形体设计了一种专用舍入方案，该方案结合了Karalias等人[NeurIPS 2025]的Carathéodory分解算法与Chekuri等人[FOCS 2010]的随机交换舍入方法。我们针对新型奖励函数证明了紧近似比，包括对数奖励函数$\varphi(x) = \log(1 + x)$的$0.827$近似比。最后，我们在合成图与真实世界图上进行了最大多重覆盖实验，表明我们的算法优于使用先进LP求解器的方法。