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 $(X,d)$ be a metric space where $X$ is a set and $d$ is a metric
A set $\mathbf {V} \subseteq {\mathcal {X}}$ , is provided together with a parameter $k$ . The goal is to find a subset ${\mathcal {C}}\subseteq \mathbf {V}$ with $|{\mathcal {C}}|=k$ such that the maximum distance of a point in $\mathbf {V}$ to the closest point in ${\mathcal {C}}$ is minimized. The problem can be formally defined as follows:
For a metric space ( ${\mathcal {X}}$ ,d),

Input: a set $\mathbf {V} \subseteq {\mathcal {X}}$ , and a parameter $k$ .
Output: a set ${\mathcal {C}}\subseteq \mathbf {V}$ of $k$ points.
Goal: Minimize the cost $r^{\mathcal {C}}(\mathbf {V} )={\underset {v\in V}{\max }}$ d(v, ${\mathcal {C}}$ )

That is, Every point in a cluster is in distance at most $r^{\mathcal {C}}(V)$ 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(v_i, v_j) ∈ N satisfying the triangle inequality, find a subset C ⊆ V with |C| = k while minimizing:

\max _{v\in V}\min _{c\in C}d(v,c)

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(e_m) and let G_i = (V, E_i), where E_i = {e₁, e₂, …, e_i}. The k-center problem is equivalent to finding the smallest index i such that G_i 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 G_i 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 ${\mathcal {C}}$ 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 ${\bar {c}}_{1}$ into $C_{1}$
For every point $v\in \mathbf {V}$ compute $d_{1}[v]$ from ${\bar {c}}_{1}$
Pick the point ${\bar {c}}_{2}$ with highest distance from ${\bar {c}}_{1}$ .
Add it to the set of centers and denote this expanded set of centers as $C_{2}$ . Continue this till k centers are found

Running time

The i^th iteration of choosing the i^th center takes ${\mathcal {O}}(n)$ time.
There are k such iterations.
Thus, overall the algorithm takes ${\mathcal {O}}(nk)$ 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 $\mathbf {V} \subseteq {\mathcal {X}}$ , belonging to a metric space ( ${\mathcal {X}}$ ,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. $r^{\mathbf {K} }(\mathbf {V} )\leq 2r^{opt}(\mathbf {V} ,{\textit {k}})$ ^[1]

This theorem can be proven using two cases as follows,

Case 1: Every cluster of ${\mathcal {C}}_{opt}$ contains exactly one point of $\mathbf {K}$

Consider a point $v\in \mathbf {V}$
Let ${\bar {c}}$ be the center it belongs to in ${\mathcal {C}}_{opt}$
Let ${\bar {k}}$ be the center of $\mathbf {K}$ that is in $\Pi ({\mathcal {C}}_{opt},{\bar {c}})$
$d(v,{\bar {c}})=d(v,{\mathcal {C}}_{opt})\leq r^{opt}(\mathbf {V} ,k)$
Similarly, $d({\bar {k}},{\bar {c}})=d({\bar {k}},{\mathcal {C}}_{opt})\leq r^{opt}$
By the triangle inequality: $d(v,{\bar {k}})\leq d(v,{\bar {c}})+d({\bar {c}},{\bar {k}})\leq 2r^{opt}$

Case 2: There are two centers ${\bar {k}}$ and ${\bar {u}}$ of $\mathbf {K}$ that are both in $\Pi ({\mathcal {C}}_{opt},{\bar {c}})$ , for some ${\bar {c}}\in {\mathcal {C}}_{opt}$ (By pigeon hole principle, this is the only other possibility)

Assume, without loss of generality, that ${\bar {u}}$ was added later to the center set $\mathbf {K}$ by the greedy algorithm, say in i^th iteration.
But since the greedy algorithm always chooses the point furthest away from the current set of centers, we have that ${\bar {k}}\in {\mathcal {C}}_{i-1}$ and,

${\begin{aligned}r^{\mathbf {K} }(\mathbf {V} )\leq r^{{\mathcal {C}}_{i-1}}(\mathbf {V} )&=d({\bar {u}},{\mathcal {C}}_{i-1})\\&\leq d({\bar {u}},{\bar {k}})\\&\leq d({\bar {u}},{\bar {c}})+d({\bar {c}},{\bar {k}})\\&\leq 2r^{opt}\end{aligned}}$ ^[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 G_i has a dominating set of size at most k and computes a maximal independent set of G_i, 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]

References

1 2 3 Har-peled, Sariel (2011). Geometric Approximation Algorithms. Boston, MA, USA: American Mathematical Society. ISBN 978-0821849118.
↑ Vazirani, Vijay V. (2003), Approximation Algorithms, Berlin: Springer, pp. 47–48, ISBN 3-540-65367-8
↑ 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
↑ 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
↑ Hochbaum, Dorit S. (1997), Approximation Algorithms for NP-Hard problems, Boston: PWS Publishing Company, pp. 346–398, ISBN 0-534-94968-1
↑ Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (2000), "Minimum k-center", A Compendium of NP Optimization Problems
↑ 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.
↑ 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.
↑ Feldmann, Andreas Emil; Marx, Dániel (2020-07-01). "The Parameterized Hardness of the k-Center Problem in Transportation Networks" (PDF). Algorithmica. 82 (7): 1989–2005. doi:10.1007/s00453-020-00683-w. ISSN 1432-0541. S2CID 3532236.
↑ 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.