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：本文研究了在单位圆盘图中检测与计数小规模模式出现的参数化复杂度问题：给定一个具有层叠度 $p$ 嵌入的 $n$ 顶点单位圆盘图 $G$（即该图表示为单位大小闭圆盘的相交图，且每个点最多被 $p$ 个圆盘包含）和一个 $k$ 顶点单位圆盘图 $P$，计算 $P$ 在 $G$ 中（诱导）拷贝的数量。针对一般模式 $P$，我们提出了一个 $2^{O(p k /\log k)}n^{O(1)}$ 时间的计数算法。我们证明该结果是紧的，即使在层叠度 $p=2$ 且 $k=n$ 的情况下：任何 $2^{o(n/\log n)}n^{O(1)}$ 时间算法都将违反指数时间假设（ETH）。对于大多数自然模式类别（如连通图和独立集），我们给出了以下结果：首先，我们提出了一个 $(pk)^{O(\sqrt{pk})}n^{O(1)}$ 时间算法，该算法在有限层叠度和多种模式下对 ETH 而言近乎最优。其次，对于 $p= k^{O(1)}$ 的情况，我们提供了图灵核化方法（即通过多项式时间预处理算法将实例规模缩减至 $k^{O(1)}$）。我们的方法融合了先前为平面子图同构开发的技术——如[Nederlof STOC'20]提出的“高效容斥原理”和[Bodlaender et al. ICALP'16]提出的“同构检验”——并结合针对单位圆盘图定制的新型分离器层次结构以及关于低阶非同构分离数的新界。