## Group-Harmonic and Group-Closeness Maximization â€“ Approximation and Engineering

Eugenio Angriman, Ruben Becker, Gianlorenzo D'Angelo, Hugo Gilbert, Alexander van der Grinten, Henning MeyerhenkeALENEX 2021

Abstract : Centrality measures characterize important nodes in networks. Efficiently computing such nodes has received a lot of attention. When considering the generalization of computing central groups of nodes, challenging optimization problems occur. In this work, we study two such problems, group-harmonic maximization and group-closeness maximization both from a theoretical and from an algorithm engineering perspective. On the theoretical side, we obtain the following results. For group-harmonic maximization, unless P = NP, there is no polynomial-time algorithm that achieves an approximation factor better than $1-\frac{1}{e}$ (directed) and $1-\frac{1}{4e}$ (undirected), even for unweighted graphs. On the positive side, we show that a greedy algorithm achieves an approximation factor of $\lambda (1-\frac{2}{e})$ (directed) and $\frac{\lambda}{2}(1-\frac{1}{e})$ (undirected), where $\lambda$ is the ratio of minimal and maximal edge weights. For group-closeness maximization, we obtain a strong separation between undirected and directed graphs (that holds even in the unweighted case). The undirected case is NP-hard to be approximated to within a factor better than $1-\frac{1}{e+1}$ and a constant approximation factor is achieved by a local-search algorithm. For the directed case, however, we show that, for any $\varepsilon <\frac{1}{2}$, the problem is NP-hard to be approximated within a factor of $4|V|^{-\varepsilon}$. From the algorithm engineering perspective, we provide efficient implementations of the above greedy and local search algorithms. In our extensive experimental study we show that, on instances small enough so that an optimum solution can be computed in reasonable time, the quality of both the greedy and the local search algorithms come very close to the optimum. On larger instances, our local search algorithms yield results with superior quality compared to existing greedy and local search solutions, at the cost of additional running time. We thus advocate local search for scenarios where solution quality is of highest concern.