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The FastMap Pipeline for Facility Location Problems

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Book cover PRIMA 2022: Principles and Practice of Multi-Agent Systems (PRIMA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13753))

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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. 1.

    Linear time after ignoring logarithmic factors.

  2. 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. 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. 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. 5.

    The ORLib instances were generated using the ORLib graphs from http://people.brunel.ac.uk/~mastjjb/jeb/orlib/pmedinfo.html.

  6. 6.

    The MovingAI instances were generated using the MovingAI graphs from https://movingai.com/benchmarks.

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Authors and Affiliations

  1. Department of Computer Science, University of Southern California, Los Angeles, USA

    Omkar Thakoor, Ang Li & Sven Koenig

  2. Information Sciences Institute, University of Southern California, Los Angeles, USA

    Srivatsan Ravi, Erik Kline & T. K. Satish Kumar

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  1. Omkar Thakoor
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  2. Ang Li
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  3. Sven Koenig
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  4. Srivatsan Ravi
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  5. Erik Kline
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  6. T. K. Satish Kumar
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Correspondence to Omkar Thakoor .

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Editors and Affiliations

  1. Özyeğin University, Istanbul, Turkey

    Reyhan Aydoğan

  2. Universitat Politècnica de València, Valencia, Spain

    Natalia Criado

  3. Université Paris-Dauphine, Paris, France

    Jérôme Lang

  4. Universitat Politècnica de València, Valencia, Spain

    Victor Sanchez-Anguix

  5. King's College London, London, UK

    Marc Serramia

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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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