We study problems related to connecting multi-interface networks of wireless devices. These problems are modeled using graphs, where vertices represent the devices and edges represent potential communication links. Each vertex can activate multiple interfaces, and a connection between two vertices is established if they share at least one common active interface. We consider two problems arising in multi-interface networks: Coverage and Connectivity. In the Coverage problem, every connection defined in the network must be established, while in the Connectivity problem, groups of terminals specified in the input should be connected. The solution should minimize the maximum cost incurred by a vertex or the total cost incurred by all vertices. In this work we are interested in approximating the former of the two cost criterions. We model both problems using ILPs and we design approximation algorithms based on a randomized rounding of the solution of the linear programming relaxation. For the Coverage problem, this yields an $O(\log m)$-approximation algorithm, which is tight, since the problem generalizes Set-Cover. This improves upon the $O(b\cdot\log n)$-approximation algorithm, where $b$ is a certain graph parameter which can be as large as $Ω(n)$ [Algorithmica '12]. The same relaxation can also be used to get an $k$-approximation algorithm, where $k$ is the number of different interfaces. This generalizes a similar result for the uniform cost case. For the Connectivity problem, we obtain an $O(\log^2 m)$-approximation algorithm, which is the first non-trivial approximation algorithm for this problem. The algorithm is based on a similar LP relaxation with additional cut constraints to ensure connectivity. The rounding procedure is similar to the one for the Coverage problem but requires a more careful analysis to ensure that the connectivity constraints are satisfied.

翻译：我们研究无线设备多接口网络的连接相关问题。这些问题通过图模型进行建模，其中顶点代表设备，边代表潜在通信链路。每个顶点可激活多个接口，当两个顶点共享至少一个公共激活接口时，它们之间即建立连接。我们考虑多接口网络中产生的两个问题：覆盖问题与连通性问题。在覆盖问题中，网络中定义的每条连接必须建立；而在连通性问题中，输入中指定的终端组应当实现连通。解决方案应最小化单个顶点产生的最大成本或所有顶点的总成本。本文关注前一种成本准则的近似方法。我们采用整数线性规划(ILP)对两个问题建模，并基于线性规划松弛解的随机舍入设计近似算法。对于覆盖问题，该算法实现了$O(\log m)$-近似比，由于该问题可归结为集合覆盖问题，此结果具有紧致性。该结果改进了此前$O(b\cdot\log n)$-近似算法（其中$b$是可达$Ω(n)$的特定图参数）[Algorithmica '12]。同一松弛模型还可导出$k$-近似算法，其中$k$为不同接口的数量，这推广了均匀成本情形的类似结论。对于连通性问题，我们获得了$O(\log^2 m)$-近似算法，这是该问题首个非平凡近似算法。该算法基于类似的线性规划松弛模型，并增加额外割约束以确保连通性。舍入过程与覆盖问题类似，但需要更精细的分析以确保连通性约束得以满足。