Camille Richer

Tom-Lukas Breitkopf, Vincent Froese, Anton Herrmann et al.

We consider the well-studied problem of finding a spanning tree with minimum average distance between vertex pairs (called a MAD tree). This is a classic network design problem which is known to be NP-hard. While approximation algorithms and polynomial-time algorithms for some graph classes are known, the parameterized complexity of the problem has not been investigated so far. We start a parameterized complexity analysis with the goal of determining the border of algorithmic tractability for the MAD tree problem. To this end, we provide a linear-time algorithm for graphs of constant modular width and a polynomial-time algorithm for graphs of bounded treewidth; the degree of the polynomial depends on the treewidth. That is, the problem is in FPT with respect to modular width and in XP with respect to treewidth. Moreover, we show it is in FPT when parameterized by vertex integrity or by an above-guarantee parameter. We complement these algorithms with NP-hardness on split graphs.

Michał Szyfelbein, Camille Richer

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.