Local Search for Group Closeness Maximization on Big Graphs

TL;DR

This paper introduces new local search heuristics for the NP-hard problem of group closeness maximization in large graphs, significantly improving speed and solution quality over existing methods.

Contribution

The authors develop scalable local search algorithms using randomized approximation and dynamic data structures for group closeness maximization.

Findings

Algorithms are 10 to 1000 times faster than greedy heuristics.

Solutions are 12-13% higher quality on weighted graphs.

Achieve near-optimal solutions within minutes on very large graphs.

Abstract

In network analysis and graph mining, closeness centrality is a popular measure to infer the importance of a vertex. Computing closeness efficiently for individual vertices received considerable attention. The NP-hard problem of group closeness maximization, in turn, is more challenging: the objective is to find a vertex group that is central as a whole and state-of-the-art heuristics for it do not scale to very big graphs yet. In this paper, we present new local search heuristics for group closeness maximization. By using randomized approximation techniques and dynamic data structures, our algorithms are often able to perform locally optimal decisions efficiently. The final result is a group with high (but not optimal) closeness centrality. We compare our algorithms to the current state-of-the-art greedy heuristic both on weighted and on unweighted real-world graphs. For graphs with hundreds of millions of edges, our local search algorithms take only around ten minutes, while greedy requires more than ten hours. Overall, our new algorithms are between one and two orders of magnitude faster, depending on the desired group size and solution quality. For example, on weighted graphs and $k = 10$, our algorithms yield solutions of $12,4\%$ higher quality, while also being $793,6\times$ faster. For unweighted graphs and $k = 10$, we achieve solutions within $99,4\%$ of the state-of-the-art quality while being $127,8\times$ faster.