The soft capacitated facility location problem (SCFLP) is a classic combinatorial optimization problem, with its variants widely applied in the fields of operations research and computer science. In the SCFLP, given a set $\mathcal{F}$ of facilities and a set $\mathcal{D}$ of clients, each facility has a capacity and an open cost, allowing to open multiple times, and each client has a demand. This problem is to find a subset of facilities in $\mathcal{F}$ and connect each client to the facilities opened, such that the total cost including open cost and connection cost is minimied. SCFLP is a NP-hard problem, which has led to a focus on approximation algorithms. Based on this, we consider a variant, that is, soft capacitated facility location problem with submodular penalties (SCFLPSP), which allows some clients not to be served by accepting the penalty cost. And we consider the integer splittable case of demand, that is, the demand of each client is served by multiple facilities with the integer service amount by each facility. Based on LP-rounding, we propose a $(\lambda R+4)$-approximation algorithm, where $R=\frac{\max_{i \in \mathcal{F} }f_i}{\min_{i \in \mathcal{F} }f_i},\lambda=\frac{R+\sqrt{R^2+8R}}{2R}$. In particular, when the open cost is uniform, the approximation ratio is 6.

翻译：软容量设施选址问题（SCFLP）是一个经典的组合优化问题，其变体在运筹学和计算机科学领域有着广泛应用。在SCFLP中，给定设施集合$\mathcal{F}$和客户集合$\mathcal{D}$，每个设施具有容量和开设成本（允许多次开设），每个客户具有需求量。该问题旨在从$\mathcal{F}$中选取一个设施子集，并将每个客户连接到已开设的设施，使得包含开设成本和连接成本的总成本最小化。SCFLP是一个NP难问题，因此近似算法成为研究重点。在此基础上，我们考虑一种变体——带次模惩罚的软容量设施选址问题（SCFLPSP），该问题允许部分客户不被服务，但需接受惩罚成本。同时我们考虑需求的整数可分割情形，即每个客户的需求可由多个设施共同满足，且每个设施提供的服务量为整数。基于LP取整方法，我们提出一个$(\lambda R+4)$-近似算法，其中$R=\frac{\max_{i \in \mathcal{F} }f_i}{\min_{i \in \mathcal{F} }f_i},\lambda=\frac{R+\sqrt{R^2+8R}}{2R}$。特别地，当开设成本相同时，近似比可达6。