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.

翻译：本文研究单位圆盘图中检测与计数小模式出现的参数化复杂度问题：给定一个具有嵌入层数$p$的$n$顶点单位圆盘图$G$（即该图表示为闭单位圆盘的相交图，且每个点至多包含于$p$个圆盘中）和一个$k$顶点单位圆盘图$P$，计算$G$中（诱导）$P$副本的数量。对于一般模式$P$，我们给出一个$2^{O(p k /\log k)}n^{O(1)}$时间算法用于计数模式出现次数。我们证明该算法紧于指数时间假设（ETH），即使对于层数$p=2$且$k=n$的情形：任何$2^{o(n/\log n)}n^{O(1)}$时间算法都将违反ETH。对于大多数自然模式类别（如连通图和独立集），我们给出以下结果：首先，我们得到一个$(pk)^{O(\sqrt{pk})}n^{O(1)}$时间算法，该算法在有界层数和多数模式情形下接近ETH下界。其次，对于$p= k^{O(1)}$的情况，我们提供图灵核化方法（即给出一个多项式时间预处理算法将实例规模缩减至$k^{O(1)}$）。我们的方法结合了先前为平面子图同构问题开发的工具，包括来自[Nederlof STOC'20]的"高效容斥原理"和来自[Bodlaender et al. ICALP'16]的"同构检测"，并引入了一种针对单位圆盘图的不同分隔符层次结构及小阶非同构分隔数量的新界。