Accelerated Relax-and-Round for Concave Coverage Problems

2026-05-07

AI Summary

This work addresses the concave coverage problem—a nonlinear generalization of the classical maximum coverage problem—where the goal is to maximize a concave reward function, such as the logarithm, under resource constraints. We propose a projected accelerated gradient method based on a smooth surrogate objective, circumventing traditional linear programming relaxations, and introduce a novel combinatorial rounding scheme tailored to the hypersimplex that integrates Carathéodory decomposition with randomized swapping. This approach yields, for the first time, a tight theoretical approximation guarantee of 0.827 for objectives like logarithmic rewards. Empirical evaluations demonstrate that the algorithm runs in $\widetilde{O}(mn \varepsilon^{-1})$ time on both synthetic and real-world graph data, significantly outperforming existing LP-based solvers.

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 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.

Problem

• concave coverage
• maximum coverage
• approximation algorithm
• combinatorial optimization

Innovation

• accelerated relax-and-round
• concave coverage
• projected accelerated gradient
• hypersimplex rounding
• approximation ratio