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.

中文翻译：

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近似由$$ \ tau $$τ松弛的软能力设施位置问题

在本文中，我们考虑了\（\ tau \）松弛的软能力设施位置问题（\（\ tau \）松弛的SCFLP），它扩展了一些众所周知的设施位置问题，例如平方度量软能力设施的位置问题（SMSCFLP），软能力设施位置问题（SCFLP），平方度量设施位置问题和无能力设施位置问题。在\（\ tau \）松弛的SCFLP中，为我们提供了一个工具集\（\ mathcal {F} \），一个客户集和一个参数\（\ tau \ ge 1 \）。每个设施\（i \ in \ mathcal {F} \）具有一个容量\（u_i \）和一个开放成本\（f_i \），并且可以多次打开。如果设施i被打开了l次，则该设施最多可以由\（l u_i \）个客户端连接，并且会产生\（l f_i \）的开放成本。每个设施-客户对都有连接成本。在连接成本为非负，对称且满足\（\ tau \）松弛三角形不等式的假设下，我们希望打开某些设施（一次或多次），并将每个客户连接到一个已打开的设施而不会违反容量限制，以使总的开放成本以及连接成本最小化。作为我们的主要贡献，我们提出了一种基于对偶的\（（3 tau + 1）\） -逼近算法\（\ tau \）-松弛的SCFLP。此外，我们的算法不仅扩展了原对偶技术的适用性，而且还将先前对SMSCFLP的近似保证从\（11.18+ \ varepsilon \）提高到了10。