While most classical NP-hard graph problems cannot be solved in time $2^{o(n)}$ on general graphs under the Exponential Time Hypothesis (ETH), many exhibit the square-root phenomenon and admit optimal algorithms running in time $2^{O(\sqrt{n})}$ on certain geometric intersection graphs, such as planar graphs or unit disk graphs. In 2018, de Berg et al. developed a general algorithmic framework for such problems on intersection graphs of similarly sized fat objects in $\mathbb{R}^d$, achieving running times of the form $2^{O(n^{1-1/d})}$, along with matching lower bounds under ETH. In this paper, we identify problems that do not exhibit the square-root phenomenon, yet still admit subexponential algorithms on intersection graphs of similarly sized fat objects in $\mathbb{R}^d$, for every fixed dimension $d \geqslant 2$. We introduce the notion of a weak square-root phenomenon: problems that can be solved in time $2^{\tilde{O}(n^{1-1/(d+1)})}$, and for which matching lower bounds hold under ETH. We develop both an algorithmic framework and a corresponding lower bound framework. As concrete examples, we show that the problems 2-Subcoloring and Two Sets Cut-Uncut exhibit this behavior. Our algorithms rely on a new win-win structural theorem, which can be informally stated as follows: every such graph admits a sublinear separator whose removal leaves connected components with sublinear independence number. To facilitate the design of these algorithms, we introduce a new graph parameter, the $α$-modulator number, which generalizes both the independence number and the vertex cover number.

翻译：尽管在指数时间假设（ETH）下，大多数经典NP难图问题在一般图上无法在$2^{o(n)}$时间内解决，但许多问题展现出平方根现象，并在某些几何交图（如平面图或单位圆盘图）上存在运行时间为$2^{O(\sqrt{n})}$的最优算法。2018年，de Berg等人针对$\mathbb{R}^d$中相似大小胖物体的交图开发了一个通用算法框架，实现了形式为$2^{O(n^{1-1/d})}$的运行时间，并在ETH下给出了匹配的下界。本文中，我们识别出那些不展现平方根现象，但仍在每个固定维度$d \geqslant 2$的$\mathbb{R}^d$中相似大小胖物体交图上存在次指数算法的问题。我们引入了弱平方根现象的概念：这类问题可在$2^{\tilde{O}(n^{1-1/(d+1)})}$时间内解决，且其在ETH下存在匹配的下界。我们同时开发了一个算法框架与相应的下界框架。作为具体实例，我们证明了2-子着色问题和二集合割-不割问题展现出这一行为。我们的算法依赖于一个新的双赢结构定理，其非正式表述如下：每个此类图都存在一个次线性分隔符，移除该分隔符后剩余连通分量的独立数均为次线性。为便于设计这些算法，我们引入了一个新的图参数——α-模数，该参数同时推广了独立数和顶点覆盖数。