Accelerated Relax-and-Round for Concave Coverage Problems

TL;DR

This paper introduces an accelerated relax-and-round algorithm for concave coverage problems, improving efficiency and approximation ratios over previous methods, with strong empirical performance.

Contribution

It replaces LP relaxation with a gradient method and develops a specialized rounding scheme, achieving faster runtime and better approximation guarantees.

Findings

Achieves $ ilde{O}(mn \, \varepsilon^{-1})$ running time.

Provides a 0.827-approximation for the logarithmic reward.

Outperforms LP-based approaches in experiments.

Abstract

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 Carath\'eodory 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.