Advances in Exact and Approximate Group Closeness Centrality Maximization

Abstract

In the NP-hard Group Closeness Centrality Maximization problem, the input is a graph $G=(V,E)$ and a positive integer $k$, and the task is to find a set $S\subseteq V$ of size $k$ that minimizes group farness $f(S)={\sum }_{v\in V}{\min }_{s\in S}\text{dist}(v,s)$. The state-of-the-art exact algorithm iteratively solves ILPs of increasing size until the final ILP can provably represent an optimal solution. We introduce a new data reduction technique that eliminates variables from the ILP by proving that certain vertices have their distance to any optimal solution structurally determined by a neighbor. Additionally, we bootstrap the exact solver with an approximate solution to produce near-sufficient ILPs from the first iteration, reducing the number of needed iterations. Our improvements yield a speedup by a factor of $4.5$ over the next best exact algorithm and can achieve speedups by up to a factor of $34.1$. Furthermore, we add reduction techniques to a $1/5$-approximation algorithm, and show that these adaptations do not compromise its approximation guarantee. The improved algorithm achieves mean speedups of up to $1.6$ and a maximum speedup of $9.6$ times. Finally, we settle an open question by proving that a widely used greedy algorithm admits arbitrarily poor approximation ratios.