带惩罚的容错设施布局问题的近似算法

在带惩罚的容错设施布局问题中, 给定顾客集合、地址集合、以及每个顾客和各个地址之间的连接费用, 这里假设连接费用是可度量的. 每位顾客有各自的服务需求, 每个地址可以开设任意多个设施, 顾客可以被安排连接到某些地址的一些开设的设施上以满足其需求, 也可以被拒绝, 但这时要支付拒绝该顾客所带来的惩罚费用. 目标是确定哪些顾客的服务需求被拒绝并开设一些设施, 将未被拒绝的顾客连接到不同的开设设施上, 使得开设费用、连接费用和惩罚费用总和最小. 给出了带惩罚的容错设施布局问题的线性整数规划及其对偶规划, 进一步, 给出了基于其线性规划和对偶规划舍入的4-近似算法.

Approximation algorithm for the fault-tolerant facility placement problem with penalties

In the fault-tolerant facility placement problem with penalties, we are given a set of customers and a set of sites, and connection costs between customers and sites. We assume that the connection costs satisfy the metric principle. Each customer has its service demands.We can open an unbounded number of different facilities at each site. Each customer can be either assigned to some different opened facilities in some sites to satisfy its demands or rejected by paying a penalty. The objective is to minimize the total cost of opening facilities and connecting those non-rejected customers todifferent open facilities, and the reject penalties. In this paper, we formulate the fault-tolerant facility placement problem with penalties as a linear integer programming and give its dual programming. Then we propose a 4-approximation algorithm based on rounding its linear programming and dual programming.