Approximating the \(\tau \)-relaxed soft capacitated facility location problem

In this paper, we consider the \(\tau \)-relaxed soft capacitated facility location problem (\(\tau \)-relaxed SCFLP), which extends several well-known facility location problems like the squared metric soft capacitated facility location problem (SMSCFLP), soft capacitated facility location problem (SCFLP), squared metric facility location problem and uncapacitated facility location problem. In the \(\tau \)-relaxed SCFLP, we are given a facility set \(\mathcal {F}\), a client set and a parameter \(\tau \ge 1\). Every facility \(i \in \mathcal {F}\) has a capacity \(u_i\) and an opening cost \(f_i\), and can be opened multiple times. If facility i is opened l times, this facility can be connected by at most \(l u_i\) clients and incurs an opening cost of \(l f_i\). Every facility-client pair has a connection cost. Under the assumption that the connection costs are non-negative, symmetric and satisfy the \(\tau \)-relaxed triangle inequality, we wish to open some facilities (once or multiple times) and connect every client to an opened facility without violating the capacity constraint so as to minimize the total opening costs as well as connection costs. As our main contribution, we propose a primal-dual based \((3 \tau + 1)\)-approximation algorithm for the \(\tau \)-relaxed SCFLP. Furthermore, our algorithm not only extends the applicability of the primal-dual technique but also improves the previous approximation guarantee for the SMSCFLP from \(11.18+ \varepsilon \) to 10.