Abstract
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 NPcomplete, 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 polytime 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(pw)n^{o(pw)} on nvertex graphs of constant degree, with 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. '17] with respect to the combined parameter tl+Delta, where tl is the treelength and Delta the maximumdegree of the input graph.
BibTeX  Entry
@InProceedings{bonnet_et_al:LIPIcs:2019:11466,
author = {{\'E}douard Bonnet and Nidhi Purohit},
title = {{Metric Dimension Parameterized by Treewidth}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {5:15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771290},
ISSN = {18688969},
year = {2019},
volume = {148},
editor = {Bart M. P. Jansen and Jan Arne Telle},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2019/11466},
URN = {urn:nbn:de:0030drops114668},
doi = {10.4230/LIPIcs.IPEC.2019.5},
annote = {Keywords: Metric Dimension, Treewidth, Parameterized Hardness}
}
Keywords: 

Metric Dimension, Treewidth, Parameterized Hardness 
Collection: 

14th International Symposium on Parameterized and Exact Computation (IPEC 2019) 
Issue Date: 

2019 
Date of publication: 

04.12.2019 