Expanding Reverse Nearest Neighbors

In a graph, the reverse nearest neighbors (RNN) of vertex f refer to the set of vertices that consider f as their nearest neighbor. When f represents a facility like a subway station, its RNN comprises potential users who prefer the nearest facility. In practice, there may be underutilized facilities with small RNN sizes, and relocating these facilities to expand their service can be costly or infeasible. A more cost-effective approach involves selectively upgrading some edges (e.g., reducing their weights) to expand the RNN sizes of underutilized facilities. This motivates our research on the Expanding Reverse Nearest Neighbors (ERNN) problem, which aims to maximize the RNN size of a target facility by upgrading a limited number of edges. Solving the ERNN problem allows underutilized facilities to serve more users and alleviate the burden on other facilities. Despite numerous potential applications, ERNN is hard to solve: It can be proven to be NP-hard and APX-hard, and it exhibits non-monotonic and non-submodular properties. To overcome these challenges, we propose novel greedy algorithms that improve efficiency by minimizing the number of edges that need to be processed and the cost of processing each edge. Experimental results demonstrate that the proposed algorithms achieve orders of magnitude speedup compared to the standard greedy algorithm while greatly expanding the RNN.