We present an evolutionary algorithm evo-SMC for the problem of Submodular Maximization under Cost constraints (SMC). Our algorithm achieves $1/2$-approximation with a high probability $1-1/n$ within $\mathcal{O}(n^2K_{\beta})$ iterations, where $K_{\beta}$ denotes the maximum size of a feasible solution set with cost constraint $\beta$. To the best of our knowledge, this is the best approximation guarantee offered by evolutionary algorithms for this problem. We further refine evo-SMC, and develop {\sc st-evo-SMC}. This stochastic version yields a significantly faster algorithm while maintaining the approximation ratio of $1/2$, with probability $1-\epsilon$. The required number of iterations reduces to $\mathcal{O}(nK_{\beta}\log{(1/\epsilon)}/p)$, where the user defined parameters $p \in (0,1]$ represents the stochasticity probability, and $\epsilon \in (0,1]$ denotes the error threshold. Finally, the empirical evaluations carried out through extensive experimentation substantiate the efficiency and effectiveness of our proposed algorithms. Our algorithms consistently outperform existing methods, producing higher-quality solutions.

翻译：我们提出了一种用于带成本约束的子模最大化问题（SMC）的进化算法 evo-SMC。该算法能在 $\mathcal{O}(n^2K_{\beta})$ 次迭代内，以高概率 $1-1/n$ 实现 $1/2$ 近似比，其中 $K_{\beta}$ 表示在成本约束 $\beta$ 下可行解集的最大规模。据我们所知，这是进化算法在该问题上所能提供的最优近似保证。我们进一步改进了 evo-SMC，并开发了{\sc st-evo-SMC}。这种随机版本在保持 $1/2$ 近似比的前提下，实现了显著更快的算法运行速度，成功概率为 $1-\epsilon$。所需迭代次数减少至 $\mathcal{O}(nK_{\beta}\log{(1/\epsilon)}/p)$，其中用户定义参数 $p \in (0,1]$ 表示随机性概率，$\epsilon \in (0,1]$ 表示误差阈值。最后，通过大量实验进行的实证评估证实了我们所提出算法的效率和有效性。我们的算法始终优于现有方法，能够产生更高质量的解决方案。