由反馈顶点集和其他结构参数参数化度量维度

Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, P. Tale
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引用次数: 1

摘要

对于图$G$,如果对于任意两个顶点$u,v \in V(G)$,存在一个顶点$w \in S$使得$d(w,u) \neq d(w,v)$,则子集$S \subseteq V(G)$称为\emph{解析集}。{\scMetric Dimension}问题以一个图$G$和一个正整数$k$作为输入,并询问是否存在一个大小不超过$k$的解析集。这个问题是在20世纪70年代提出的,已知是\NP -hard [Garey和Johnson的书中的GT 61]。在参数化复杂度领域,Hartung和Nichterlein [CCC 2013]证明了当用自然参数$k$参数化时问题是\W[2] -hard。他们还观察到,当用顶点覆盖数参数化时,它是\FPT,并询问了它在\emph{较小}参数下的复杂性,特别是反馈顶点集数。我们通过证明用参数反馈顶点集数加路径宽度的组合参数化{\sc度量维度}是\W[1] -hard来回答这个问题。这也改进了Bonnet和Purohit [IPEC 2019]的结果,该结果指出问题是\W[1] -硬参数化的路径宽度。从积极的方面来看,我们表明,当用到簇的距离或到共簇的距离参数化时,{\sc度量维度}是\FPT,这两个参数都比顶点覆盖数小。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters
For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set} if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ such that $d(w,u) \neq d(w,v)$. The {\sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be \NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is \W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is \FPT\ when parameterized by the vertex cover number and asked about its complexity under \emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {\sc Metric Dimension} is \W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is \W[1]-hard parameterized by the pathwidth. On the positive side, we show that {\sc Metric Dimension} is \FPT\ when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.
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