arXiv 2103.055882021-03cs.CCcs.DS

退化图中的精确与近似模式计数：新算法、硬度结果和复杂性二分法

Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies

Marco Bressan, Marc Roth

备注：: 44 pages, 3 figures

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摘要：

研究了$d$ -退化$n$-顶点图$G$中的$k$ -顶点图$H$的同态计数、副本计数和诱导副本计数问题.我们的主要结果建立了详尽和明确的复杂性分类计数子图和诱导子图。我们发现，$G$中的$H$（不一定是诱导的）拷贝可以在时间$f(k,d)\cdot n^{\max(\mathsf{imn}(H),1)}\cdot \log n$中计数，其中$f$是一些可计算的函数，$\mathsf{imn}(H)$是$H$的最大诱导匹配的大小。只要允许模式类具有无界的诱导匹配数，该算法本质上是最优的：除非指数时间假设（ETH）失败，否则对于任何函数$f$，没有算法在时间上运行$f(k,d)\cdot n^{o(\mathsf{imn}(H)/\log \mathsf{imn}(H))}$。在计数诱导子图的情况下，我们得到了一个类似的分类：我们可以在时间$f(k,d)\cdot n^{α(H)}\cdot \log n$上计数$G$中的诱导子图$H$的拷贝数，如果允许模式类具有无界的独立数，则在时间$f(k,d)\cdot n^{o(α(H)/\log α(H))}$上运行的算法是不可能的，除非ETH失败.在语言的参数化的复杂性，我们的结果产生二分法在固定参数易处理和$\#\mathsf{W}[1]$ -硬的情况下，如果我们参数化的模式和退化的主机图的大小。我们的结果意味着，除非ETH失败，否则几种模式不能在时间$f(k,d)\cdot n^{o(k/\log k)}$中计数，包括$k$ -匹配，$k$ -独立集，（诱导）$k$ -路径，（诱导）$k$ -圈，和诱导$(k,k)$ -双列集.这些精确计数的下限补充了新的算法，近似计数的子图和诱导子图的退化图。

英文摘要：

We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a $k$-vertex graph $H$ in a $d$-degenerate $n$-vertex graph $G$. Our main result establishes exhaustive and explicit complexity classifications for counting subgraphs and induced subgraphs. We show that the (not necessarily induced) copies of $H$ in $G$ can be counted in time $f(k,d)\cdot n^{\max(\mathsf{imn}(H),1)}\cdot \log n$, where $f$ is some computable function and $\mathsf{imn}(H)$ is the size of the largest induced matching of $H$. Whenever the class of allowed patterns has unbounded induced matching number, this algorithm is essentially optimal: Unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm running in time $f(k,d)\cdot n^{o(\mathsf{imn}(H)/\log \mathsf{imn}(H))}$ for any function $f$. In case of counting induced subgraphs, we obtain a similar classification along the independence number $α$: we can count the induced copies of $H$ in $G$ in time $f(k,d)\cdot n^{α(H)}\cdot \log n$, and if the class of allowed patterns has unbounded independence number, an algorithm running in time $f(k,d)\cdot n^{o(α(H)/\log α(H))}$ is impossible, unless ETH fails. In the language of parameterized complexity, our results yield dichotomies in fixed-parameter tractable and $\#\mathsf{W}[1]$-hard cases if we parameterize by the size of the pattern and the degeneracy of the host graph. Our results imply that several patterns cannot be counted in time $f(k,d)\cdot n^{o(k/\log k)}$, including $k$-matchings, $k$-independent sets, (induced) $k$-paths, (induced) $k$-cycles, and induced $(k,k)$-bicliques, unless ETH fails. Those lower bounds for exact counting are complemented with new algorithms for approximate counting of subgraphs and induced subgraphs in degenerate graphs.

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摘要：

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