🤖 AI Summary
This paper studies robust subexponential algorithms for the Hamiltonian Path/Cycle and Parameterized Long Path problems on geometric intersection graphs of similarly sized “fat” objects in ℝᵈ. We introduce the novel tool of λ-connected partitions and establish a low-treewidth pattern-covering theorem that is independent of concrete geometric representations, unifying divide-and-conquer with structural graph analysis. Our approach yields the first ETH-tight robust algorithms: Hamiltonian Path/Cycle is solved in 2ᴼ(ⁿ¹⁻¹⁄ᵈ) time, and the k-Path problem in 2ᴼ(ᵏ¹⁻¹⁄ᵈlog²k) ⋅ nᴼ(¹) time. These results overcome longstanding geometric-dependence barriers, significantly advancing the theoretical frontier of parameterized algorithms on geometric graphs.
📝 Abstract
We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in $mathbb{R}^d$. In this setting, each vertex corresponds to a geometric object, and two vertices are adjacent if and only if their objects intersect. We introduce a new tool for designing such algorithms, which we call a $lambda$-linked partition. This is a partition of the vertex set into groups of highly connected vertices. Crucially, such a partition can be computed in polynomial time and does not require access to the geometric representation of the graph. We apply this framework to problems related to paths and cycles in graphs. First, we obtain the first robust ETH-tight algorithms for Hamiltonian Path and Hamiltonian Cycle, running in time $2^{O(n^{1-1/d})}$ on intersection graphs of similarly sized fat objects in $mathbb{R}^d$. This resolves an open problem of de Berg et al. [STOC 2018] and completes the study of these problems on geometric intersection graphs from the viewpoint of ETH-tight exact algorithms. We further extend our approach to the parameterized setting and design the first robust subexponential parameterized algorithm for Long Path in any fixed dimension $d$. More precisely, we obtain a randomized robust algorithm running in time $2^{O(k^{1-1/d}log^2 k)}, n^{O(1)}$ on intersection graphs of similarly sized fat objects in $mathbb{R}^d$, where $k$ is the natural parameter. Besides $lambda$-linked partitions, our algorithm also relies on a low-treewidth pattern covering theorem that we establish for geometric intersection graphs, which may be viewed as a refinement of a result of Marx-Pilipczuk [ESA 2017]. This structural result may be of independent interest.