Subexponential algorithms in geometric graphs via the subquadratic grid minor property: the role of local radius

TL;DR

The paper addresses subexponential parameterized algorithms for Triangle Hitting, Feedback Vertex Set, and Odd Cycle Transversal in geometric intersection graphs, focusing on how subquadratic and almost subquadratic grid-minor properties and local radius influence tractability. It introduces an ASQGM framework, extends it to local-radius based bounds, and shows that string-related geometric graph classes (including square and contact-segment graphs) admit subexponential FPT algorithms for these problems under suitable parameters, while providing ETH-based hardness results that delineate the limits of these techniques. The main contributions include explicit subexponential running times for FVS in contact segment graphs, TH in $K_{t,t}$-free $d$-DIR graphs, and TH in contact segment graphs, together with general theorems that connect ASQGM properties, neighborhood complexity, and local radius to algorithmic efficiency. Collectively, the results demonstrate that subexponential parameterized algorithms in geometric graphs require additional structural restrictions and highlight the critical role of local radius in extending bidimensional techniques to broader graph classes with practical geometric representations.

Abstract

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.

Figures (9)

• Figure 1: Left: a representation of a disk graph. Right: a contact 2-DIR graph and the corresponding graph. In these graphs (where the left one is from fomin2018excluded), $\omega(G)$ is constant, $\mathop{\mathrm{\textsf{tw}}}\nolimits(G) \ge t$ (where $t=3$ here) as it contains $K_{t,t}$ as a minor, and $\mathop{\mathrm{\boxplus}}\nolimits(G) = \mathcal{O}(\sqrt{t})$ as they have a feedback vertex set of size at most $t$.
• Figure 2: Inclusion between graph classes.
• Figure 3: From left to right, four representations of contact string graphs, then a representation of 3-DIR contact-segment graph, and finally on the right an example of an intersection between segments not allowed in a representation of a contact-segment graph.
• Figure 4: A graphical representation of the definition of $\mathcal{G}^{(c_1,c_2)}$, adapted from baste2022contraction.
• Figure 6: Illustrations of the construction used in the proof of the \ref{['lm:curve']}. Squares of $I(v)$ are represented in green. Top left: construction used for the \ref{['cl:diag']} . Top right: construction used for the \ref{['cl:intersect']}. Bottom left: construction used for \ref{['cl:goingaround']}. Observe that in this situation $c_a$ and $c_b$ are next to opposite sides of the square containing $c_0$, that $\mathcal{C}_a^*$ can be extended in an counterclockwise direction, and $\mathcal{C}_b^*$ in a clockwise direction, which ensure the existence of a common point $c$ of their monotonic extensions. Bottom right: an example of a $4$-monotonic curve between $a$ and $b$ obtained by the construction of \ref{['lm:curve']}. Observe that only two squares of $I(v)$ are crossed.

Theorems & Definitions (40)

• Definition 1: baste2022contraction
• Definition 2
• Theorem 3: Theorem 1.1 in lokSODA22
• Corollary 4: Corollary 1.1 in lokSODA22 restricted to FVS
• Definition 5
• Claim 6
• Remark 7
• Lemma 8: Adapted from lok2021complete
• Lemma 9: Adapted from lok2021complete
• Corollary 10