In the generalized budgeted submodular set function maximization problem, we are given a ground set of elements and a set of bins. Each bin has its own cost and the cost of each element depends on its associated bin. The goal is to find a subset of elements along with an associated set of bins such that the overall costs of both is at most a given budget, and the profit is maximized. We present an algorithm that guarantees a $\frac{1}{2}(1-\frac{1}{e^{\alpha }})$-approximation, where $\alpha \le 1$ is the approximation factor of an algorithm for a sub-problem. If the costs satisfy a specific condition, we provide a polynomial-time algorithm that gives us $\alpha =1-ϵ$, while for the general case we design an algorithm with $\alpha =1-\frac{1}{e}-ϵ$.

We extend our results providing a bi-criterion approximation algorithm where we can spend an extra budget up to a factor $\beta \ge 1$ to guarantee a $\frac{1}{2}(1-\frac{1}{e^{\alpha \beta }})$-approximation.

在广义预算子模集函数最大化问题中，给定了一组基本元素和一组 bin。每个 bin 都有自己的成本，每个元素的成本取决于其关联的 bin。目标是找到一个元素子集以及一组相关联的 bin，使得两者的总成本最多是给定的预算，并且利润最大化。我们提出了一种算法来保证$\frac{1}{2}(1-\frac{1}{{\mathrm{电子}}^{\alpha }})$-近似，其中 $\alpha \le 1$是子问题算法的逼近因子。如果成本满足特定条件，我们提供多项式时间算法$\alpha =1-\epsilon$，而对于一般情况，我们设计了一个算法 $\alpha =1-\frac{1}{\mathrm{电子}}-\epsilon$.

我们扩展了我们的结果，提供了一个双准则近似算法，我们可以在其中花费一个额外的预算 $\beta \ge 1$ 保证一个 $\frac{1}{2}(1-\frac{1}{{\mathrm{电子}}^{\alpha \beta }})$-近似。