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 $k$).

翻译：我们研究“至少k个顶点的最密子图”（DALkS）问题，其中给定一个无向图$G$和一个整数$k$，目标是寻找$G$中具有至少$k$个顶点且密度最大的子图。已知最优算法由Khuller与Saha（2009）以及Andersen（2007）独立发现，可在多项式时间内对DALkS实现2-近似。本文给出一个（简单的）从“k个顶点的最密子图”（DkS）到“至少k个顶点的最密子图”的归约，表明：若DkS难以在任何常数因子内近似，则对于任意$\varepsilon > 0$，DALkS难以在$(3/2 - \varepsilon)$因子内近似。该结论同时适用于标准（非参数化）设定和参数化（以$k$为参数）设定。随后我们将该归约推广，以证明“至少k个顶点的最密子图”在$(2 - \varepsilon)$因子内近似具有紧的下界困难性——尽管需要基于一个更强的假设，该假设大致表述为：对任意常数$\delta> 0$，DkS难以在$k^{1 - \delta}$因子内近似。此结论同样自然延伸至参数化设定。此前，DALkS的$(2 - \varepsilon)$因子不可近似性仅在小集扩张假设（Bergner, 2013; Manurangsi, 2017）下成立，而该假设不适用于问题的参数化版本。此外，我们证明DALkS的精确版本是W[1]-困难的（以$k$为参数）。