Algorithms and Turing Kernels for Detecting and Counting Small Patterns in Unit Disk Graphs
Jesper Nederlof, Krisztina Szilágyi
TL;DR
The work tackles the problem of detecting and counting small pattern copies in unit disk graphs, a setting where geometry imposes novel structural constraints. It develops subexponential parameterized algorithms that count subgraph and induced-subgraph occurrences for a $k$-vertex pattern $P$ in an $n$-vertex host $G$, with running time $(pk)^{O(\,sqrt{pk})}\,\sigma_{O(\,sqrt{pk})}(P)^2 imes n^{O(1)}$. The approach combines dynamic programming over non-isomorphic separations, efficient inclusion-exclusion, and a separator hierarchy tailored to unit disk graphs, and it includes a Turing kernelization that reduces the host to a $(O(k) imes O(k))$-box with ply $p$. ETH-based lower bounds show tightness in several regimes, while near-optimal results are achieved for natural pattern classes such as connected graphs or independent sets. These techniques extend the reach of subgraph isomorphism methods to geometric intersection graphs and point to broader applicability in disk-graph and higher-dimensional settings.
Abstract
In this paper we investigate the parameterized complexity of the task of counting and detecting occurrences of small patterns in unit disk graphs: Given an $n$-vertex unit disk graph $G$ with an embedding of ply $p$ (that is, the graph is represented as intersection graph with closed disks of unit size, and each point is contained in at most $p$ disks) and a $k$-vertex unit disk graph $P$, count the number of (induced) copies of $P$ in $G$. For general patterns $P$, we give an $2^{O(p k /\log k)}n^{O(1)}$ time algorithm for counting pattern occurrences. We show this is tight, even for ply $p=2$ and $k=n$: any $2^{o(n/\log n)}n^{O(1)}$ time algorithm violates the Exponential Time Hypothesis (ETH). For most natural classes of patterns, such as connected graphs and independent sets we present the following results: First, we give an $(pk)^{O(\sqrt{pk})}n^{O(1)}$ time algorithm, which is nearly tight under the ETH for bounded ply and many patterns. Second, for $p= k^{O(1)}$ we provide a Turing kernelization (i.e. we give a polynomial time preprocessing algorithm to reduce the instance size to $k^{O(1)}$). Our approach combines previous tools developed for planar subgraph isomorphism such as `efficient inclusion-exclusion' from [Nederlof STOC'20], and `isomorphisms checks' from [Bodlaender et al. ICALP'16] with a different separator hierarchy and a new bound on the number of non-isomorphic separations of small order tailored for unit disk graphs.