Polylogarithmic Approximation for Covering and Connecting Multi-Interface Networks
We study problems related to connecting multi-interface networks of wireless devices. These problems can be 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. However, activating an interface induces a cost that depends both on the type of the interface and on the vertex that activates it. 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, it is only required that the established connections form a subgraph spanning the network. The solution should also minimize the maximum cost incurred by a node or the total cost incurred by all vertices. We model both problems using Integer Linear Programming (ILP) 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 $O(\log n)$-approximation algorithm, where $n$ is the number of vertices. This result 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 main result of our work is an $O(\log^2 n)$-approximation algorithm for the Connectivity, which is the first non-trivial approximation for this problem. The algorithm is based on a similar LP relaxation with additional cut constraints to ensure connectivity. The rounding procedure resembles the one for the Coverage but requires a more careful analysis to ensure that the connectivity constraints are satisfied.