Addressing the Hardness of k-Facility Relocation Problem: A Pair of Approximate Solutions

Facility Relocation (FR), which is an effort to reallocate the placement of facilities to adapt to the changes of urban planning and population distribution, has remarkable impact on many application areas. Existing solutions to the FR problem either focus on relocating one facility (i.e., 1-FR) or fail to guarantee the result quality on relocating k > 1 facilities (i.e., k-FR). As k-FR problem is NP-hard and is not submodular or non-decreasing, traditional hill-climb approximate algorithm cannot be directly applied. In light of that, we propose to transform k-FR into another facility placement problem, which is submodular and non-decreasing. We theoretically prove that the optimal solution of both problems are equivalent. Accordingly, we are able to present the first approximate solution towards the k-FR, namely FR2FP. Our extensive comparison over both FR2FP and the state-of-the-art heuristic solution shows that FR2FP, although provides approximation guarantee, cannot necessarily given superior results to the heuristic solution. The comparison motivates and, more importantly, directs us to present an advanced approximate solution, namely FR2FP-ex. Extensive experimental study over both real-world and synthetic datasets have verified that, FR2FP-ex demonstrates the best result quality. In addition, we also exactly unveil the scenarios when the state-of-the-art heuristic would fail to provide satisfied results in practice.