Maximum Betweenness Centrality: Approximability and Tractable Cases

The Maximum Betweenness Centrality problem (MBC) can be defined as follows. Given a graph find a $k$-element node set $C$ that maximizes the probability of detecting communication between a pair of nodes $s$ and $t$ chosen uniformly at random. It is assumed that the communication between $s$ and $t$ is realized along a shortest $s$--$t$ path which is, again, selected uniformly at random. The communication is detected if the communication path contains a node of $C$. Recently, Dolev et al. (2009) showed that MBC is NP-hard and gave a $(1-1/e)$-approximation using a greedy approach. We provide a reduction of MBC to Maximum Coverage that simplifies the analysis of the algorithm of Dolev et al. considerably. Our reduction allows us to obtain a new algorithm with the same approximation ratio for a (generalized) budgeted version of MBC. We provide tight examples showing that the analyses of both algorithms are best possible. Moreover, we prove that MBC is APX-complete and provide an exact polynomial-time algorithm for MBC on tree graphs.