Metric k-center

In graph theory, the metric k-center problem is a combinatorial optimization problem studied in theoretical computer science. Given n cities with specified distances, one wants to build k warehouses in different cities and minimize the maximum distance of a city to a warehouse. In graph theory, this means finding a set of k vertices for which the largest distance of any point to its closest vertex in the k-set is minimum. The vertices must be in a metric space, providing a complete graph that satisfies the triangle inequality.

Formal definition

Let be a metric space where is a set and is a metric
A set , is provided together with a parameter . The goal is to find a subset with such that the maximum distance of a point in to the closest point in is minimized. The problem can be formally defined as follows:
For a metric space (,d),

  • Input: a set , and a parameter .
  • Output: a set of points.
  • Goal: Minimize the cost d(v,)

That is, Every point in a cluster is in distance at most from its respective center. [1]

The k-Center Clustering problem can also be defined on a complete undirected graph G = (V, E) as follows:
Given a complete undirected graph G = (V, E) with distances d(vi, vj)  N satisfying the triangle inequality, find a subset C  V with |C| = k while minimizing:

Computational complexity

In a complete undirected graph G = (V, E), if we sort the edges in non-decreasing order of the distances: d(e1)  d(e2)    d(em) and let Gi = (V, Ei), where Ei = {e1, e2, , ei}. The k-center problem is equivalent to finding the smallest index i such that Gi has a dominating set of size at most k. [2]

Although Dominating Set is NP-complete, the k-center problem remains NP-hard. This is clear, since the optimality of a given feasible solution for the k-center problem can be determined through the Dominating Set reduction only if we know in first place the size of the optimal solution (i.e. the smallest index i such that Gi has a dominating set of size at most k), which is precisely the difficult core of the NP-Hard problems. Although a Turing reduction can get around this issue by trying all values of k.

Approximations

A simple greedy algorithm

A simple greedy approximation algorithm that achieves an approximation factor of 2 builds using a farthest-first traversal in k iterations. This algorithm simply chooses the point farthest away from the current set of centers in each iteration as the new center. It can be described as follows:

  • Pick an arbitrary point into
  • For every point compute from
  • Pick the point with highest distance from .
  • Add it to the set of centers and denote this expanded set of centers as . Continue this till k centers are found

Running time

  • The ith iteration of choosing the ith center takes time.
  • There are k such iterations.
  • Thus, overall the algorithm takes time.[3]

Proving the approximation factor

The solution obtained using the simple greedy algorithm is a 2-approximation to the optimal solution. This section focuses on proving this approximation factor.

Given a set of n points , belonging to a metric space (,d), the greedy K-center algorithm computes a set K of k centers, such that K is a 2-approximation to the optimal k-center clustering of V.

i.e. [1]

This theorem can be proven using two cases as follows,

Case 1: Every cluster of contains exactly one point of

  • Consider a point
  • Let be the center it belongs to in
  • Let be the center of that is in
  • Similarly,
  • By the triangle inequality:


Case 2: There are two centers and of that are both in , for some (By pigeon hole principle, this is the only other possibility)

  • Assume, without loss of generality, that was added later to the center set by the greedy algorithm, say in ith iteration.
  • But since the greedy algorithm always chooses the point furthest away from the current set of centers, we have that and,

[1]

Another 2-factor approximation algorithm

Another algorithm with the same approximation factor takes advantage of the fact that the k-Center problem is equivalent to finding the smallest index i such that Gi has a dominating set of size at most k and computes a maximal independent set of Gi, looking for the smallest index i that has a maximal independent set with a size of at least k. [4] It is not possible to find an approximation algorithm with an approximation factor of 2  ε for any ε > 0, unless P = NP. [5] Furthermore, the distances of all edges in G must satisfy the triangle inequality if the k-center problem is to be approximated within any constant factor, unless P = NP. [6]

Parameterized approximations

It can be shown that the k-Center problem is W[2]-hard to approximate within a factor of 2  ε for any ε > 0, when using k as the parameter.[7] This is also true when parameterizing by the doubling dimension (in fact the dimension of a Manhattan metric), unless P=NP.[8] When considering the combined parameter given by k and the doubling dimension, k-Center is still W[1]-hard but it is possible to obtain a parameterized approximation scheme.[9] This is even possible for the variant with vertex capacities, which bound how many vertices can be assigned to an opened center of the solution.[10]

See also

References

  1. Har-peled, Sariel (2011). Geometric Approximation Algorithms. Boston, MA, USA: American Mathematical Society. ISBN 978-0821849118.
  2. Vazirani, Vijay V. (2003), Approximation Algorithms, Berlin: Springer, pp. 47–48, ISBN 3-540-65367-8
  3. Gonzalez, Teofilo F. (1985), "Clustering to minimize the maximum intercluster distance", Theoretical Computer Science, vol. 38, Elsevier Science B.V., pp. 293–306, doi:10.1016/0304-3975(85)90224-5
  4. Hochbaum, Dorit S.; Shmoys, David B. (1986), "A unified approach to approximation algorithms for bottleneck problems", Journal of the ACM, vol. 33, pp. 533–550, doi:10.1145/5925.5933, ISSN 0004-5411, S2CID 17975253
  5. Hochbaum, Dorit S. (1997), Approximation Algorithms for NP-Hard problems, Boston: PWS Publishing Company, pp. 346–398, ISBN 0-534-94968-1
  6. Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (2000), "Minimum k-center", A Compendium of NP Optimization Problems
  7. Feldmann, Andreas Emil (2019-03-01). "Fixed-Parameter Approximations for k-Center Problems in Low Highway Dimension Graphs" (PDF). Algorithmica. 81 (3): 1031–1052. doi:10.1007/s00453-018-0455-0. ISSN 1432-0541. S2CID 46886829.
  8. Feder, Tomás; Greene, Daniel (1988-01-01). "Optimal algorithms for approximate clustering". Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88. New York, NY, USA: Association for Computing Machinery. pp. 434–444. doi:10.1145/62212.62255. ISBN 978-0-89791-264-8. S2CID 658151.
  9. Feldmann, Andreas Emil; Marx, Dániel (2020-07-01). "The Parameterized Hardness of the k-Center Problem in Transportation Networks". Algorithmica. 82 (7): 1989–2005. doi:10.1007/s00453-020-00683-w. ISSN 1432-0541. S2CID 3532236.
  10. Feldmann, Andreas Emil; Vu, Tung Anh (2022). "Generalized $$k$$-Center: Distinguishing Doubling and Highway Dimension". In Bekos, Michael A.; Kaufmann, Michael (eds.). Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science. Vol. 13453. Cham: Springer International Publishing. pp. 215–229. arXiv:2209.00675. doi:10.1007/978-3-031-15914-5_16. ISBN 978-3-031-15914-5.

Further reading

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