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A rounding and clustering-based exact algorithm for the p-center problem

Zacharie Ales, Cristian Duran-Matelunaa, Sourour Elloumi

TL;DR

The paper tackles the NP-hard $p$-center problem by introducing a scalable exact algorithm that tightly couples client clustering with iterative distance rounding. It initializes a compact representative set via $k$-means medoids, then solves $(PCP)_\alpha$ on progressively finer distance grids, updating the representative set through cluster quadrants and local search while pruning dominated sites. The core contributions are the distance-rounding scheme that limits Distinct Distances to at most 10 per rounded instance, the cluster-quadrant based update of representatives, and domination-based pruning, all validated on very large-scale instances up to $1.9$ million clients/sites. Empirically, the method outperforms two state-of-the-art exact approaches for $p>5$ in terms of final gap and runtime, highlighting strong practical impact for large-scale facility location problems. These results demonstrate a meaningful advance in exact PCP resolution, enabling exact solutions to substantially larger instances than previous methods.

Abstract

The p-center problem consists in selecting p facilities from a set of possible sites and allocating a set of clients to them in such a way that the maximum distance between a client and the facility to which it is allocated is minimized. This paper proposes a new scalable exact solution algorithm based on client clustering and an iterative distance rounding procedure. The client clustering enables to initialize and update a subset of clients for which the p-center problem is iteratively solved. The rounding drastically reduces the number of distinct distances considered at each iteration. Our algorithm is tested on 396 benchmark instances with up to 1.9 million clients and facilities. We outperform the two state-of-the-art exact methods considered when p is not very small (i.e., p > 5).

A rounding and clustering-based exact algorithm for the p-center problem

TL;DR

The paper tackles the NP-hard -center problem by introducing a scalable exact algorithm that tightly couples client clustering with iterative distance rounding. It initializes a compact representative set via -means medoids, then solves on progressively finer distance grids, updating the representative set through cluster quadrants and local search while pruning dominated sites. The core contributions are the distance-rounding scheme that limits Distinct Distances to at most 10 per rounded instance, the cluster-quadrant based update of representatives, and domination-based pruning, all validated on very large-scale instances up to million clients/sites. Empirically, the method outperforms two state-of-the-art exact approaches for in terms of final gap and runtime, highlighting strong practical impact for large-scale facility location problems. These results demonstrate a meaningful advance in exact PCP resolution, enabling exact solutions to substantially larger instances than previous methods.

Abstract

The p-center problem consists in selecting p facilities from a set of possible sites and allocating a set of clients to them in such a way that the maximum distance between a client and the facility to which it is allocated is minimized. This paper proposes a new scalable exact solution algorithm based on client clustering and an iterative distance rounding procedure. The client clustering enables to initialize and update a subset of clients for which the p-center problem is iteratively solved. The rounding drastically reduces the number of distinct distances considered at each iteration. Our algorithm is tested on 396 benchmark instances with up to 1.9 million clients and facilities. We outperform the two state-of-the-art exact methods considered when p is not very small (i.e., p > 5).

Paper Structure

This paper contains 12 sections, 1 theorem, 1 equation, 4 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. An exact rounding and clustering-based algorithm for (PCP)
  4. The initialization steps
  5. Exact solution of the -center problem for the rounding precision
  6. Updating set through the clusters quadrants and local search
  7. Improving the computation of the sites dominations
  8. An illustrative example
  9. Computational experiments
  10. Features comparison of our method
  11. Comparison to state-of-the-art methods
  12. Conclusions

Key Result

Proposition 1

There exists an optimal solution of $(PCP_{\mathcal{R}})$ in which no dominated site is opened.

Figures (4)

  • Figure 1: Example of an update of a set $\mathcal{R}$ and three clusters.
  • Figure 2: Illustration of our algorithm on instance nu3496 with $p=5$. The medoids of the clusters are represented by stars, the other clients in the representative set by triangles and the sites opened in the optimal solution of each iteration by red squares.
  • Figure 3: Performance profiles of our algorithm with different features deactivated.
  • Figure 4: Performance profiles of our algorithm compared with two state-of-the-art methods.

Theorems & Definitions (3)

  • Definition 1: church1984
  • Proposition 1: church1984
  • Definition 2: Distance rounding