稀疏图中多支配与支配模式的细粒度复杂度
Fine-Grained Complexity of Multiple Domination and Dominating Patterns in Sparse Graphs
AI总结：
本研究针对稀疏图中的支配问题变体，推导了r-多重k-支配集和三类H-支配集的条件最优算法运行时间，证明其相比经典支配集问题可实现更大加速。
摘要：
图中支配问题的研究已催生了文献中多种不同的支配问题。其中大多数遵循如下通用框架：给定图$G$和整数$k$，判断是否存在大小为$k$的顶点集$S$，使得(1)满足某种内部性质$ϕ(S)$（例如连通性），且(2)每个顶点$v$满足某种支配性质$ρ(S, v)$（例如存在$s∈S$与$v$相邻）。\n 由于许多现实世界的图都是稀疏图，我们旨在确定这类问题关于图$G$的顶点数$n$和边数$m$的最优运行时间。虽然经典支配集问题在稀疏图中的改进相当有限（Fischer, Künnemann, Redzic SODA'24），但我们证明文献中研究的自然变体可实现大得多的加速，且存在多种可能的运行时间。具体而言，我们得到了以下问题的条件最优算法：\n 1) $r$-多重$k$-支配集（每个顶点必须至少与$S$中的$r$个顶点相邻）：若$r≤k-2$，我们得到运行时间为$(m/n)^{r} n^{k-r+o(1)}$的算法，在3-均匀超团假设下该算法是条件最优的。在稀疏图中，该运行时间根据$r$的不同在$n^{k-1±o(1)}$和$n^{2±o(1)}$之间完全插值。值得注意的是，当$r=k-1$时，我们得到一种随机算法，其性能优于$(m/n)^{k-1} n^{1+o(1)}$，且我们证明该算法在$k$-团假设下接近最优。\n 2) $H$-支配集（$S$必须诱导出模式$H$）。我们条件性地解决了三个此类问题的复杂度：(a) 支配团（Dominating Clique，$H$为$k$-团），(b) 大小为$k$的极大独立集（Maximal Independent Set of size $k$，$H$为$k$个顶点的独立集），(c) 支配诱导匹配（Dominating Induced Matching，$H$为$k$个顶点的完美匹配）。
英文摘要：
The study of domination in graphs has led to a variety of domination problems studied in the literature. Most of these follow the following general framework: Given a graph $G$ and an integer $k$, decide if there is a set $S$ of $k$ vertices such that (1) some inner property $ϕ(S)$ (e.g., connectedness) is satisfied, and (2) each vertex $v$ satisfies some domination property $ρ(S, v)$ (e.g., there is an $s\in S$ that is adjacent to $v$). Since many real-world graphs are sparse, we seek to determine the optimal running time of such problems in both the number $n$ of vertices and the number $m$ of edges in $G$. While the classic dominating set problem admits a rather limited improvement in sparse graphs (Fischer, Künnemann, Redzic SODA'24), we show that natural variants studied in the literature admit much larger speed-ups, with a diverse set of possible running times. Specifically, we obtain conditionally optimal algorithms for: 1) $r$-Multiple $k$-Dominating Set (each vertex must be adjacent to at least $r$ vertices in $S$): If $r\le k-2$, we obtain a running time of $(m/n)^{r} n^{k-r+o(1)}$ that is conditionally optimal assuming the 3-uniform hyperclique hypothesis. In sparse graphs, this fully interpolates between $n^{k-1\pm o(1)}$ and $n^{2\pm o(1)}$, depending on $r$. Curiously, when $r=k-1$, we obtain a randomized algorithm beating $(m/n)^{k-1} n^{1+o(1)}$ and we show that this algorithm is close to optimal under the $k$-clique hypothesis. 2) $H$-Dominating Set ($S$ must induce a pattern $H$). We conditionally settle the complexity of three such problems: (a) Dominating Clique ($H$ is a $k$-clique), (b) Maximal Independent Set of size $k$ ($H$ is an independent set on $k$ vertices), (c) Dominating Induced Matching ($H$ is a perfect matching on $k$ vertices).
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