对偶参数化的图模体问题
Graph Motif Problems Parameterized by Dual
摘要：
设$G=(V,E)$是顶点着色图，其中$C$是用于对$V$着色的颜色集合。图基序（或GM）问题将$G$（从$C$构建的颜色的多集$M$）作为输入，并询问是否存在子集$S\subseteq V$，使得（i）$G[S]$是连接的，以及（ii）从$S$获得的颜色的多集等于$M$。彩色图基序（CGM）问题是GM的特殊情况，其中$M$是一个集合，列表彩色图基序（LGM）问题是GM的扩展，其中$V$的每个顶点$v$可以从颜色列表$\mathcal{L}(v)\subseteq C$中选择其颜色。我们研究了由对偶参数$\ell:=|V|-|M|$参数化的GM、CGM和LGM三个问题。对于一般图，我们表明，假设强指数时间假设，CGM没有$(2-ε)^\ell\cdot |V|^{\mathcal{O}(1)}$时间算法，这意味着在$\mathcal{O}(2^\ell\cdot |E|)$时间内运行的先前算法是最佳的[Betzler等人，IEEE/ACM TCBB 2011]。我们还证明了LGM对于$\ell$是W[1]-困难的，即使我们限制自己至多两个颜色的列表。如果我们限制输入图是一棵树，那么我们表明，GM可以解决在$\mathcal{O}(3^\ell\cdot |V|)$时间，但不承认多项式大小的问题的核心，而CGM可以解决在$\mathcal{O}(\sqrt{2}^{\ell} + |V|)$时间，并承认一个多项式大小的问题的核心。
英文摘要：
Let $G=(V,E)$ be a vertex-colored graph, where $C$ is the set of colors used to color $V$. The Graph Motif (or GM) problem takes as input $G$, a multiset $M$ of colors built from $C$, and asks whether there is a subset $S\subseteq V$ such that (i) $G[S]$ is connected and (ii) the multiset of colors obtained from $S$ equals $M$. The Colorful Graph Motif (or CGM) problem is the special case of GM in which $M$ is a set, and the List-Colored Graph Motif (or LGM) problem is the extension of GM in which each vertex $v$ of $V$ may choose its color from a list $\mathcal{L}(v)\subseteq C$ of colors. We study the three problems GM, CGM, and LGM, parameterized by the dual parameter $\ell:=|V|-|M|$. For general graphs, we show that, assuming the strong exponential time hypothesis, CGM has no $(2-ε)^\ell\cdot |V|^{\mathcal{O}(1)}$-time algorithm, which implies that a previous algorithm, running in $\mathcal{O}(2^\ell\cdot |E|)$ time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that LGM is W[1]-hard with respect to $\ell$ even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that GM can be solved in $\mathcal{O}(3^\ell\cdot |V|)$ time but admits no polynomial-size problem kernel, while CGM can be solved in $\mathcal{O}(\sqrt{2}^{\ell} + |V|)$ time and admits a polynomial-size problem kernel.
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