Subexponential Algorithms in Geometric Graphs via the Subquadratic Grid Minor Property: The Role of Local Radius

Abstract

We investigate the existence in geometric graph classes of subexponential parameterized algorithms for cycle-hitting problems like Triangle Hitting (TH), Feedback Vertex Set (FVS) or Odd Cycle Transversal (OCT). These problems respectively ask for the existence in a graph G of a set X of at most k vertices such that G-X is triangle-free, acyclic, or bipartite. It is know that subexponential FPT algorithms of the form 2^o(k)n^𝒪(1) exist in planar and even H-minor free graphs from bidimensionality theory [Demaine et al. 2005], and there is a recent line of work lifting these results to geometric graph classes consisting of intersection of similarly sized "fat" objects ([Fomin et al. 2012], [Grigoriev et al. 2014], or disk graphs [Lokshtanov et al. 2022], [An et al. 2023]). 
In this paper we first identify sufficient conditions, for any graph class 𝒞 included in string graphs, to admit subexponential FPT algorithms for any problem in 𝒫, a family of bidimensional problems where one has to find a set of size at most k hitting a fixed family of graphs, containing in particular FVS. Informally, these conditions boil down to the fact that for any G ∈ 𝒞, the local radius of G (a new parameter introduced in [Lokshtanov et al. 2023]) is polynomial in the clique number of G and in the maximum matching in the neighborhood of a vertex. To demonstrate the applicability of this generic result, we bound the local radius for two special classes: intersection graphs of axis-parallel squares and of contact graphs of segments in the plane. This implies that any problem Π ∈ 𝒫 (in particular, FVS) can be solved in:  
- 2^𝒪(k^{3/4}log k) n^𝒪(1)-time in contact segment graphs, 
- 2^𝒪(k^{9/10}log k) n^𝒪(1) in intersection graphs of axis-parallel squares  On the positive side, we also provide positive results for TH by solving it in:  
- 2^𝒪(k^{3/4}log k) n^𝒪(1)-time in contact segment graphs, 
- 2^𝒪(√dt²(log t)k^{2/3}log k) n^𝒪(1)-time in K_{t,t}-free d-DIR graphs (intersection of segments with d slopes) 
On the negative side, assuming the ETH we rule out the existence of algorithms solving:  
- TH and OCT in time 2^o(n) in 2-DIR graphs and more generally in time 2^o(√{Δn}) in 2-DIR graphs with maximum degree Δ, and 
- TH, FVS, and OCT in time 2^o(√n) in K_{2,2}-free contact-2-DIR graphs of maximum degree 6.
Observe that together, these results show that the absence of large K_{t,t} is a necessary and sufficient condition for the existence of subexponential FPT algorithms for TH in 2-DIR.