Abstract
Facility Location Problems (FLPs) involve the placement of facilities in a shared environment for serving multiple customers while minimizing transportation and other costs. FLPs defined on graphs are very general and broadly applicable. Two such fundamental FLPs are the Vertex K-Center (VKC) and the Vertex K-Median (VKM) problems. Although both these problems are NP-hard, many heuristic and approximation algorithms have been developed for solving them in practice. However, state-of-the-art heuristic algorithms require the input graph G to be complete, in which the edge joining two vertices is also the shortest path between them. When G doesn’t satisfy this property, these heuristic algorithms have to be invoked only after computing the metric closure of G, which in turn requires the computation of all-pairs shortest-path (APSP) distances. Existing APSP algorithms, such as the Floyd-Warshall algorithm, have a poor time complexity, making APSP computations a bottleneck for deploying the heuristic algorithms on large VKC and VKM instances. To remedy this, we propose the use of a novel algorithmic pipeline based on a graph embedding algorithm called FastMap. FastMap is a near-linear-time algorithm that embeds the vertices of G in a Euclidean space while approximately preserving the shortest-path distances as Euclidean distances for all pairs of vertices. The FastMap embedding can be used to circumvent the barrier of APSP computations, creating a very efficient pipeline for solving FLPs. On the empirical front, we provide test results that demonstrate the efficiency and effectiveness of our novel approach.
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Notes
- 1.
Linear time after ignoring logarithmic factors.
- 2.
The edit distance between two strings is the minimum number of insertions, deletions, or substitutions that are needed to transform one to the other.
- 3.
Unless \(|E| = O(|V|)\), in which case the complexity is near-linear in the size of the input because of the \(\log |V|\) factor.
- 4.
The DIMACS instances were generated using the DIMACS graphs from http://networkrepository.com/dimacs.php and https://mat.tepper.cmu.edu/COLOR/instances.html.
- 5.
The ORLib instances were generated using the ORLib graphs from http://people.brunel.ac.uk/~mastjjb/jeb/orlib/pmedinfo.html.
- 6.
The MovingAI instances were generated using the MovingAI graphs from https://movingai.com/benchmarks.
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Thakoor, O., Li, A., Koenig, S., Ravi, S., Kline, E., Satish Kumar, T.K. (2023). The FastMap Pipeline for Facility Location Problems. In: Aydoğan, R., Criado, N., Lang, J., Sanchez-Anguix, V., Serramia, M. (eds) PRIMA 2022: Principles and Practice of Multi-Agent Systems. PRIMA 2022. Lecture Notes in Computer Science(), vol 13753. Springer, Cham. https://doi.org/10.1007/978-3-031-21203-1_25
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