The Hardness and Approximation of the Densest k-Subgraph Problem in Parameterized Metric Graphs

A complete weighted graph G = (V, E, w) is called Δ
_β
-metric, for some β ≥ 1/2, if G satisfies the β-triangle inequality, i.e., w(u, v) ≤ β · (w(u, x) + w(x, v)) for all vertices u, v, x ∈ V . Given a Δ
_β
-metric graph G = (V, E, w), the Δ
_β
-WEIGHTED DENSEST k-SUBGRAPH (Δ
_β
-WDkS) problem is to find an induced subgraph G[C] with exactly k vertices such that the total edge weight of G[C] is maximized. For β = 1, this problem, Δ-WDkS, is known NP-hard and admits a 1/2-approximation algorithms. In this paper, we show that for any β > 1/2, Δ
_β
-WDkS is NP-hard. We also show how to modify any α-approximation algorithm for Δ-WDkS to obtain a δ
_α,β
-approximation algorithm for Δ
_β
-WDkS with δ
_α,β
> α for every β <; 1. Moreover, we prove that Δ
_β
-WDkS can be approximated to within a factor 1/2β for any β 1/2.