AI 中文总结
本文研究树上的批量购买型设施选址问题（BBFL）及相关k-电缆设施选址问题（kCFL），针对不同需求变体给出近似算法或难度结论，为树实例提供了近似比保证。
AI 中文摘要
我们研究批量购买型设施选址问题（BBFL），该问题结合了经典设施选址问题与批量购买型网络设计，其研究动机源于电信网络。问题设定为：给定带边长度、开设成本及各顶点需求的图，以及一个单调次可加的容量-成本函数，任务是在部分顶点上开设设施，并将每个顶点的需求路由至这些设施。我们要最小化解的总成本，该成本包括所选设施的开设成本，加上每条边的成本，边成本等于其长度乘以根据容量-成本函数为边中通过的需求提供足够容量的成本。该问题的常见变体为k-电缆设施选址问题（kCFL），该变体中容量通过购买给定类型的电缆副本提供，每种电缆具有特定容量和成本。我们研究树实例上的BBFL，针对单位需求和可拆分变体，证明该问题存在多项式时间近似方案（PTAS），即对任意ε>0，存在(1+ε)近似算法。我们还考虑了电缆不可拆分需求的新设定下的kCFL，该设定中顶点的需求无法在多条电缆间拆分。我们证明该问题在星形图上近似比优于3/2是NP难的，随后为树实例提供了一种算法，该算法输出的解成本最优，但每条电缆的容量超出量为1+ε倍。由此，我们证明该问题在树实例上存在2近似算法。
英文摘要
We consider the buy-at-bulk facility location problem (BBFL), a problem combining the classic facility location problem with buy-at-bulk network design, which finds motivation in telecommunication networks. In it, we are given a graph with edge lengths, opening costs and demands for each vertex, and a monotone and subadditive capacity-cost function, and our task is to open facilities on a subset of the vertices and route the demand from each vertex to these facilities. The cost of a solution (which we want to minimize) is given by the opening costs of the chosen facilities, plus the cost on each edge, which is given by its length times the cost of providing enough capacity for the demands through the edge, given by the capacity-cost function. A common variant of the problem, the $k$-cable facility location problem (kCFL), considers the case where capacity is provided by buying copies of given cable types, each with a certain capacity and cost. We study BBFL on tree instances and show, for the unit-demand and splittable variants, that the problem admits a PTAS (a $(1+ε)$-approximation for any $ε> 0$). We also consider kCFL in the new setting of cable-unsplittable demands, where the demand of a vertex cannot be split among multiple cables. We show that the problem is NP-hard to approximate to a factor better than $3/2$ on stars, and then provide an algorithm for tree instances that outputs a solution with optimal cost, but which exceeds the capacity on each cable by a factor of $1+ε$. As a consequence, we show that the problem has a $2$-approximation algorithm on trees.
CommentsPresented at WAOA 2025
DOI：10.1007/978-3-032-06706-7_12
