P-Center

[math]\displaystyle{ p }[/math]-Center --- [math]\displaystyle{ p }[/math]-центр.

A [math]\displaystyle{ p }[/math]-center of [math]\displaystyle{ G = (V,E) }[/math] is a set [math]\displaystyle{ C \subseteq V }[/math] that realizes the [math]\displaystyle{ p }[/math]-radius of [math]\displaystyle{ G }[/math].

The [math]\displaystyle{ p }[/math]-center problem is one of the main problems in facility location theory. For general graphs, the problem of finding [math]\displaystyle{ p }[/math]-centers is NPhard. Polynomial algorithms for the [math]\displaystyle{ p }[/math]-center problem are known only for trees and almost-trees. The best known algorithms have time complexity [math]\displaystyle{ {\mathcal O}(|V|) }[/math] for unweighted trees and [math]\displaystyle{ {\mathcal O}(|V|\cdot \log^{2}|V|) }[/math] for weighted trees.