Metric k-center

In graph theory, the metric k-center or metric facility location 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, or in other words a complete graph that satisfies the triangle inequality.

Formal definition

Given a complete undirected graph G = (V, E) with distances d(v_i, v_j) ∈ N satisfying the triangle inequality, find a subset S ⊆ V with |S| = k while minimizing:

$\max_{v \in V} \min_{s \in S} d(v,s)$

Computational complexity

If we sort the edges in nondecreasing order of the distances: d(e₁) ≤ d(e₂) ≤ … ≤ d(e_m) and let G_i = (V_i, 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. 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.

Approximations

A simple greedy approximation algorithm that achieves an approximation factor of 2 builds S using a farthest-first traversal in k iterations. The first iteration chooses an arbitrary vertex and adds it to S. Each subsequent iteration chooses a vertex v for which d(S, v) is maximized and adds v to S. The running time of the algorithm is O(nk).

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 on the of G_i, looking for the smallest index i that has a maximal independent set with a size of at least k.

It is not possible to find an approximation algorithm with an approximation factor of 2 − ε for any ε > 0, unless P = NP. Furthermore, the distances of all edges in G must satisfy the triangle inequality if the k-center problem is to be approximated unless P = NP.

See also

• Traveling salesman problem
• Minimum k-cut
• Dominating set
• Independent set (graph theory)
• Facility location problem