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：本文研究了几何图类中三个基本环击中问题的次指数参数化算法存在性。所考虑的问题包括：三角形击中（TH）、反馈顶点集（FVS）和奇环横贯（OCT），分别要求在图$G$中存在一个大小不超过$k$的顶点集$X$，使得$G-X$为无三角形图、无环图或二分图。根据二维性理论，已知在平面图甚至$H$-无minor图中存在此类次指数参数化算法[Demaine等，JACM 2005]，且近期有一系列工作将这些结果推广到由"胖"物体交集构成的几何图类（[Grigoriev等，FOCS 2022]和[Lokshtanov等，SODA 2022]）。本文通过考虑平面中具有$d$种可能斜率的线段交集图（$d$-DIR图）以及平面中线段接触图，聚焦于"瘦"物体。在假设指数时间假设（ETH）成立的前提下，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的算法；- 在$K_{2,2}$-自由接触2-DIR图中以$2^{o(\sqrt{n})}$时间求解TH、FVS和OCT的算法。这些结果表明，为获得这些问题的次指数参数化算法需要施加额外限制。在此方向上，我们提出：- 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$K_{t,t}$-自由$d$-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。