Optimal Hub Labeling is NP-complete

TL;DR

This paper proves that finding the optimal hub labeling for distance queries in large networks is NP-hard by reducing from the Vertex Cover problem, confirming the computational difficulty of the problem.

Contribution

It provides the first proof that the problem of assigning minimal hub labels in distance labeling is NP-hard, resolving a long-standing conjecture.

Findings

Optimal hub labeling problem is NP-hard.

Reduces from Vertex Cover problem to prove NP-hardness.

Establishes computational complexity of hub labeling.

Abstract

Distance labeling is a preprocessing technique introduced by Peleg [Journal of Graph Theory, 33(3)] to speed up distance queries in large networks. Herein, each vertex receives a (short) label and, the distance between two vertices can be inferred from their two labels. One such preprocessing problem occurs in the hub labeling algorithm [Abraham et al., SODA'10]: the label of a vertex v is a set of vertices x (the "hubs") with their distance d(x,v) to v and the distance between any two vertices u and v is the sum of their distances to a common hub. The problem of assigning as few such hubs as possible was conjectured to be NP-hard, but no proof was known to date. We give a reduction from the well-known Vertex Cover problem on graphs to prove that finding an optimal hub labeling is indeed NP-hard.