毛虫带宽复杂度的参数化及加权路径仿真
Parameterized complexity of Bandwidth of Caterpillars and Weighted Path Emulation
摘要：
在本文中，我们表明，带宽是很难的复杂性类$W[t]$的所有$t\in {\bf N}$，即使毛毛虫的头发长度最多为3。作为中间问题，我们引入了加权路径仿真问题：给定一条顶点加权路径$P_N$和整数$M$，判断是否存在$P_N$的顶点到路径$P_M$的映射，使得相邻顶点映射到相邻或相等的顶点，并且使得来自$P_M$的顶点的图像的总权重等于整数$c$。证明了以$c$为参数的{ \sc加权路径仿真}对所有$t\in {\bf N}$都是$W[t]$困难的，并且是强NP完全的.我们还表明，定向带宽是很难$W[t]$的所有$t\in {\bf N}$，有向无环图的基础无向图是一个毛毛虫。
英文摘要：
In this paper, we show that Bandwidth is hard for the complexity class $W[t]$ for all $t\in {\bf N}$, even for caterpillars with hair length at most three. As intermediate problem, we introduce the Weighted Path Emulation problem: given a vertex-weighted path $P_N$ and integer $M$, decide if there exists a mapping of the vertices of $P_N$ to a path $P_M$, such that adjacent vertices are mapped to adjacent or equal vertices, and such that the total weight of the image of a vertex from $P_M$ equals an integer $c$. We show that {\sc Weighted Path Emulation}, with $c$ as parameter, is hard for $W[t]$ for all $t\in {\bf N}$, and is strongly NP-complete. We also show that Directed Bandwidth is hard for $W[t]$ for all $t\in {\bf N}$, for directed acyclic graphs whose underlying undirected graph is a caterpillar.
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