Fast approximation algorithms for the 1-median problem on real-world large graphs

The 1-median problem (1MP) on undirected weighted graphs seeks to find a facility location minimizing the total weighted distance to all customer nodes. Although the 1MP can be solved exactly by computing the single-source shortest paths from each customer node, such approaches become computationally expensive on large-scale graphs with millions of nodes. In many real-world applications, such as recommendation systems based on large-scale knowledge graphs, the number of nodes (i.e., potential facility locations) is enormous, whereas the number of customer nodes is relatively small and spatially concentrated. In such cases, exhaustive graph exploration is not only inefficient but also unnecessary. Leveraging this observation, we propose three approximation algorithms that reduce computation by terminating Dijkstra's algorithm early. We provide theoretical analysis showing that one of the proposed algorithms guarantees an approximation ratio of 2, whereas the other two improve this ratio to 1.618. We demonstrate that the lower bound of the approximation ratio is 1.2 by presenting a specific instance. Moreover, we show that all proposed algorithms return optimal solutions when the number of customer nodes is less than or equal to three. Extensive experiments demonstrate that our algorithms significantly outperform baseline exact methods in runtime while maintaining near-optimal accuracy across all tested graph types. Notably, on grid graphs with 10 million nodes, our algorithms obtains all optimal solutions within 1 millisecond, whereas the baseline exact method requires over 70 seconds on average.