Robust Algorithms for Path and Cycle Problems in Geometric Intersection Graphs

TL;DR

This work develops robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects, achieving ETH-tight bounds for Hamiltonian Path and Cycle in 2^{O(n^{1-1/d})} time and a robust subexponential FPT algorithm for Long Path in 2^{O(k^{1-1/d} log^2 k)} n^{O(1)} time. The core technique is a λ-linked partition, a geometry-free partition into highly connected parts whose contraction has bounded degree and sublinear treewidth, enabling reductions and dynamic programming on low-treewidth graphs. A low-treewidth pattern covering theorem further strengthens the framework, providing a probabilistic method to capture relevant vertex sets while preserving structural compactness. The results extend the robustness framework of earlier work and introduce new tools with potential independent interest in geometric graph theory, while leaving open questions about derandomization and Long Cycle robustness.

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 $λ$-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 $λ$-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.

Figures (3)

• Figure 1: An example of 3 fat objects, the blue disks representing the enclosing disks of each object. The objects on left and right intersect the same object so there are at distance at most $2\beta$.
• Figure 2: Illustration of the proof of Theorem \ref{['thm:main1']}, when turning a Hamiltonian cycle C in H (left) to a Hamiltonian cycle in G (right), focusing on a part $V_i$. In both figures, the grey box represents the set $V_i$, and the blue box represents the (partial) Hamiltonian cycle going through $V_i$. Left: $C$ uses blue edges to enter and leave $V_i$, and uses edges from $G$ as well as red edges inside $V_i$ (notice that we only represented the red edges which are used by $C$). Right: since $G[V_i]$ is Hamiltonian-$\lambda$-linked, the $(s_i, t_i)$-paths which were using red edges in $H$ can be replaced by actual paths in $G$.
• Figure 3: Illustration of the five graphs of the proof of Theorem \ref{['thm:Main2']}.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Definition 3: kappa-partition de2018framework
• Definition 4: $\mathcal{P}$-contraction de2018framework
• Theorem 5: de Berg et al. de2018framework
• Remark 6
• Lemma 7: fomin2024path
• Lemma 8: Corollary of Lemma \ref{['lemma:TMembedding']}
• Definition 9: $\lambda$-linked partition
• Theorem 10