Metric Dimension Parameterized By Treewidth

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 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 $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ 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 $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

Metric dimension parameterized by treewidth in chordal graphs

“…Metric Dimension is FPT paramerized by the modular width [BFGR16]. Using Courcelle's theorem, one can also remark that it is FPT paramerized by the treedepth of the graph as observed in [GHK + 22].…”

“…Foucaud et al proved that it is FPT parameterized by the solution size in interval graphs in [FMN + 17]. This result was extended by Belmonte et al who proved in [BFGR16] that Metric Dimension is FPT parameterized by the size of the solution plus the tree-length of the graph. In particular, it implies that computing the metric dimension for chordal graph is FPT parameterized by the size of the solution.…”

Quasi-Ultrafilter on the Connectivity System: Its Relationship to Branch-Decomposition

“…The definition of branch-decomposition is shown below. Due to its significance, branch-decomposition has been the subject of various research studies [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

The Parameterized Complexity of Network Microaggregation

“…A core ingredient for the proof is that if two vertices have the same type and are both in the past or both in the future, then they are "indistinguishable" from the viewpoint of the bag. We remark that the notion of center-types is related to the well-studied METRIC DIMENSION problem (Bonnet and Purohit 2021).…”