A set of vertices W in a graph G is called resolving if for any two distinct $x,y\in V(G)$, there is $v\in W$ such that ${\mathrm{d}}_G(v,x)\ne {\mathrm{d}}_G(v,y)$, where ${\mathrm{d}}_G(u,v)$ denotes the length of a shortest path between u and v in the graph G. The metric dimension $\mathrm{md}(G)$ of G is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether $\mathrm{md}(G)⩽k$, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by k was proved to be $\mathsf{W}[2]$-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether $\mathrm{md}(G)⩽n-k$, where n is the order of G. We prove that the dual parameterization admits (a) a kernel with at most $6(k+1)$ vertices and (b) a randomized algorithm of runtime $O^{⁎}(4^{k+o(k)})$. Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number $vc(G)$ of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by $vc(G)+k$, unless NP ⊆ coNP/poly. Our reduction also gives evidence for non-existence of polynomial Turing kernels. We also prove that Metric Dimension parameterized by bandwidth or cutwidth does not admit a polynomial kernel, unless NP ⊆ coNP/poly. Finally, using Eppstein's results (2015) we show that Metric Dimension parameterized by max-leaf number does admit a polynomial kernel.

如果对于任意两个不同的点，则图G中的一组顶点W称为解析$X，ÿ\in V（G）$， 有 $v\in \mathrm{w\; ^}$ 这样 ${\mathrm{d}}_G（v，X）\ne {\mathrm{d}}_G（v，ÿ）$，在哪里 ${\mathrm{d}}_G（ü，v）$表示图形G中u和v之间最短路径的长度。公制尺寸$\mathrm{md}（G）$的ģ是拆分组的最小基数。在公制尺寸问题，即决定是否$\mathrm{md}（G）⩽ķ$甚至对于间隔图也是NP完全的（Foucaud et al。，2017）。我们从参数化复杂度的角度研究度量维度（对于任意图）。由k参数化的问题被证明是$\mathsf{w\; ^}[2]$-Hardung和Nichterlein（2013）提出的问题，我们研究了双重参数化，即是否 $\mathrm{md}（G）⩽ñ-ķ$，其中n是G的阶数。我们证明对偶参数化允许（a）内核最多$6（ķ+\mathrm{1个}）$ 顶点和（b）运行时的随机算法 ${Ø}^{⁎}（4^{ķ+Ø（ķ）}）$。Hartung和Nichterlein（2013）还观察到，通过顶点覆盖数进行参数化时，公制尺寸是固定参数可处理的$vC（G）$输入图的 我们通过证明即使参数为，也不允许多项式核来补充该观察结果$vC（G）+ķ$，除非NP⊆coNP / poly。我们的减少也提供了不存在多项式图灵核的证据。我们还证明，除非NP⊆coNP / poly，否则用带宽或cutwidth参数化的Metric Dimension不允许多项式内核。最后，使用Eppstein的结果（2015年），我们证明了通过最大叶数参数化的Metric Dimension确实接受了多项式核。