In this paper we investigate the existence of subexponential parameterized algorithms of three fundamental cycle-hitting problems in geometric graph classes. The considered problems, \textsc{Triangle Hitting} (TH), \textsc{Feedback Vertex Set} (FVS), and \textsc{Odd Cycle Transversal} (OCT) ask for the existence in a graph $G$ of a set $X$ of at most $k$ vertices such that $G-X$ is, respectively, triangle-free, acyclic, or bipartite. Such subexponential parameterized algorithms are known to exist in planar and even $H$-minor free graphs from bidimensionality theory [Demaine et al., JACM 2005], and there is a recent line of work lifting these results to geometric graph classes consisting of intersection of "fat" objects ([Grigoriev et al., FOCS 2022] and [Lokshtanov et al., SODA 2022]). In this paper we focus on "thin" objects by considering intersection graphs of segments in the plane with $d$ possible slopes ($d$-DIR graphs) and contact graphs of segments in the plane. Assuming the ETH, we rule out the existence of algorithms: - solving TH in time $2^{o(n)}$ in 2-DIR graphs; and - solving TH, FVS, and OCT in time $2^{o(\sqrt{n})}$ in $K_{2,2}$-free contact 2-DIR graphs. These results indicate that additional restrictions are necessary in order to obtain subexponential parameterized algorithms for %these problems. In this direction we provide: - a $2^{O(k^{3/4}\cdot \log k)}n^{O(1)}$-time algorithm for FVS in contact segment graphs; - a $2^{O(\sqrt d\cdot t^2 \log t\cdot k^{2/3}\log k)} n^{O(1)}$-time algorithm for TH in $K_{t,t}$-free $d$-DIR graphs; and - a $2^{O(k^{7/9}\log^{3/2}k)} n^{O(1)}$-time algorithm for TH in contact segment graphs.

翻译：本文研究几何图类中三个基本环击中问题是否存在亚指数参数化算法。所考虑的问题包括\textsc{三角形击中}(TH)、\textsc{反馈顶点集}(FVS)和\textsc{奇环横贯}(OCT)，它们询问图$G$中是否存在一个大小至多为$k$的顶点集$X$，使得$G-X$分别满足无三角形、无环或二分图性质。根据二维性理论，已知在平面图乃至$H$-minor-free图中存在此类亚指数参数化算法[Demaine等, JACM 2005]，并且近期有一系列工作将这些结果推广到由“胖”对象相交构成的几何图类([Grigoriev等, FOCS 2022]和[Lokshtanov等, SODA 2022])。本文聚焦于“瘦”对象，考虑平面上具有$d$种可能斜率的线段相交图($d$-DIR图)以及平面上线段接触图。在指数时间假设(ETH)下，我们排除了以下算法的存在性： - 在2-DIR图中以$2^{o(n)}$时间求解TH； - 在$K_{2,2}$-free接触2-DIR图中以$2^{o(\sqrt{n})}$时间求解TH、FVS和OCT。 这些结果表明，要获得这些问题的亚指数参数化算法需要附加限制条件。在此方向上我们给出： - 针对接触线段图中FVS的$2^{O(k^{3/4}\cdot \log k)}n^{O(1)}$时间算法； - 针对$K_{t,t}$-free $d$-DIR图中TH的$2^{O(\sqrt d\cdot t^2 \log t\cdot k^{2/3}\log k)} n^{O(1)}$时间算法； - 针对接触线段图中TH的$2^{O(k^{7/9}\log^{3/2}k)} n^{O(1)}$时间算法。