Subset selection with cost constraints aims to select a subset from a ground set to maximize a monotone objective function without exceeding a given budget, which has various applications such as influence maximization and maximum coverage. In real-world scenarios, the budget, representing available resources, may change over time, which requires that algorithms must adapt quickly to new budgets. However, in this dynamic environment, previous algorithms either lack theoretical guarantees or require a long running time. The state-of-the-art algorithm, POMC, is a Pareto optimization approach designed for static problems, lacking consideration for dynamic problems. In this paper, we propose BPODC, enhancing POMC with biased selection and warm-up strategies tailored for dynamic environments. We focus on the ability of BPODC to leverage existing computational results while adapting to budget changes. We prove that BPODC can maintain the best known $(\alpha_f/2)(1-e^{-\alpha_f})$-approximation guarantee when the budget changes. Experiments on influence maximization and maximum coverage show that BPODC adapts more effectively and rapidly to budget changes, with a running time that is less than that of the static greedy algorithm.

翻译：具有成本约束的子集选择问题旨在从基础集合中选出一个子集，在不超过给定预算的条件下最大化一个单调目标函数，该问题在影响力最大化与最大覆盖等场景中具有广泛应用。在实际应用中，代表可用资源的预算可能随时间变化，这就要求算法必须快速适应新的预算条件。然而，在这种动态环境中，现有算法要么缺乏理论保证，要么需要较长的运行时间。当前最优算法POMC是一种针对静态问题设计的帕累托优化方法，未考虑动态问题。本文提出BPODC算法，通过引入针对动态环境设计的偏置选择与预热策略对POMC进行改进。我们重点关注BPODC在适应预算变化时利用已有计算结果的能力。理论证明表明，当预算发生变化时，BPODC能够保持目前最优的$(\alpha_f/2)(1-e^{-\alpha_f})$近似比保证。在影响力最大化与最大覆盖问题上的实验表明，BPODC能更高效、更快速地适应预算变化，其运行时间少于静态贪心算法。