A Note on Approximability of Densest At-Least-k-Subgraph

2026-05-25

AI Summary

This work investigates the approximability of the Densest At-Least-$k$-Subgraph (DALkS) problem. By constructing reductions from the Densest $k$-Subgraph (DkS) problem, it establishes strong inapproximability bounds within both classical and parameterized complexity frameworks. The main contributions include proving, for the first time in the parameterized setting, that DALkS admits no $(2-\varepsilon)$-approximation algorithm unless DkS is efficiently approximable, and showing that its exact version is W[1]-hard. Furthermore, the paper establishes approximation hardness of factors $(3/2-\varepsilon)$ in the general setting and $(2-\varepsilon)$ in the parameterized setting—significantly strengthening prior results based on the Small Set Expansion Hypothesis.

Abstract

We study the Densest At-Least-$k$-Subgraph (DAL$k$S) problem, in which we are given an undirected graph $G$ and an integer $k$, and the goal is to find a subgraph of $G$ with at least $k$ vertices with maximum density. The best-known algorithm, independently discovered by Khuller and Saha (2009) and by Andersen (2007), yields a 2-approximation for DAL$k$S in polynomial time. In this note, we provide a (simple) reduction from Densest $k$-Subgraph (D$k$S) to Densest At-Least-$k$-Subgraph, which shows that, if D$k$S is hard to approximate to within any constant factor, then DAL$k$S is hard to approximate to within $(3/2 - \varepsilon)$ factor for every $\varepsilon > 0$. This holds in both the normal (non-parameterized) and the parameterized (by $k$) settings. We then generalize the reduction to provide a tight $(2 - \varepsilon)$ factor hardness of approximating Densest At-Least-$k$-Subgraph, albeit under a stronger hypothesis which roughly states that Densest $k$-Subgraph is hard to approximate to within $k^{1 - δ}$ factor for any constant $δ> 0$. Once again, this extends naturally to the parameterized setting. Previously, $(2 - \varepsilon)$ factor inapproximability for DAL$k$S was only known under the Small Set Expansion Hypothesis (Bergner, 2013; Manurangsi, 2017), which does not apply to the parameterized version of the problem. Furthermore, we show that the exact version of DAL$k$S is W[1]-hard (parameterized by $k$).

Problem

• Densest At-Least-k-Subgraph
• approximability
• hardness of approximation
• parameterized complexity
• Densest k-Subgraph

Innovation

• approximation hardness
• Densest At-Least-k-Subgraph
• parameterized complexity
• W[1]-hardness
• reduction