We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time $n^{\Omega(\log{n})}$ (so that the state-of-the-art running time of $n^{O(\log n)}$ is optimal up to a constant in the exponent). We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter $k$: Densest $k$-Subgraph, Smallest $k$-Edge Subgraph, Densest $k$-Subhypergraph, Steiner $k$-Forest, and Directed Steiner Network with $k$ terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves $o(k)$-approximation for Densest $k$-Subgraph. This inapproximability ratio improves upon the previous best $k^{o(1)}$ factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter $k$. Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, which improves the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis.

中文翻译：

---

强植集团假说及其后果

我们制定了一个新的硬度假设，即Strongish Planted Clique Hypothesis (SPCH)，它假设任何种植集团的算法都必须及时运行 $n^{\Omega(\log{n})}$（这样状态$n^{O(\log n)}$ 的最佳运行时间在指数中是一个常数）。我们提供了新假设的两组应用。首先，我们证明 SPCH 对以下经过充分研究的问题在参数 $k$ 方面（几乎）隐含了严格的不可逼近性结果：Densest $k$-Subgraph、Smallest $k$-Edge Subgraph、Densest $k$-Subhypergraph 、Steiner $k$-Forest 和具有 $k$ 终端对的定向 Steiner 网络。例如，我们表明，在 SPCH 下，没有多项式时间算法可以实现 Densest $k$-Subgraph 的 $o(k)$-近似。这种不可逼近率改进了之前最好的 $k^{o(1)}$ 因子（Chalermsook 等人，FOCS 2017）。此外，我们的下界甚至可以与参数 $k$ 的固定参数易处理算法相抗衡。我们的第二个应用程序侧重于图形模式检测的复杂性。对于诱导和非诱导图形模式检测，我们证明了 SPCH 下的硬度结果，这提高了 (Dalirrooyfard 等人，STOC 2019) 在指数时间假设下获得的运行时间下限。