A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time \(f(\text {pw})n^{o(\text {pw})}\) on n-vertex graphs of constant degree, with \(\text {pw}\) the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter \(\text {tl}+\Delta\), where \(\text {tl}\) is the tree-length and \(\Delta\) the maximum-degree of the input graph.

图G的分解集S是其顶点的子集，因此G的两个顶点都不具有与S相同的距离矢量。在公制尺寸这道题的解决设定的最低规模，并在其决定的形式，顶多有些规定整数解析集的大小。这个问题是NP完全的，并且在非常有限的图类中仍然存在。关于解的大小，它也是W [2]-完全的。公制尺寸已证明在有界树宽图上难以捉摸。在算法方面，对于树，甚至对于外平面图，都知道多项式时间算法，但是树宽的一般情况是最多两个。在复杂性方面，没有参数化的硬度是已知的。这导致有关该主题的几篇论文要求就树宽而言，度量维度的参数化复杂性。我们提供该问题的第一个答案。我们表明，由输入图的树宽参数化的度量维是W [1] -hard。精细化我们证明了，除非指数时间设定失败，没有算法求解公制尺寸的时间\（F（\ {文字PW}）N ^ {O（\ {文字PW}）} \）的ñ恒定度-点图形，用\（\ {文本PW} \）的输入图的pathwidth，和˚F任何可计算函数。这与Belmonte等人的FPT算法形成鲜明对比。（SIAM J Discrete Math 31（2）：1217–1243，2017）关于组合参数\（\ text {tl} + \ Delta \），其中\（\ text {tl} \）是树长和\（\ Delta \）输入图的最大度数。