Hardness of Metric Dimension in Graphs of Constant Treewidth

TL;DR

This paper proves that the Metric Dimension problem is NP-hard even in graphs with a fixed treewidth of 24, resolving a long-standing open question about its computational complexity in such graph classes.

Contribution

It establishes NP-hardness of Metric Dimension in graphs of constant treewidth, strengthening previous W[1]-hardness results and closing the complexity gap.

Findings

NP-hardness in graphs of treewidth 24

Strengthens previous W[1]-hardness results

Clarifies complexity landscape of Metric Dimension

Abstract

The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph $G$. Here, a set $S \subseteq V(G)$ is resolving if no two distinct vertices of $G$ have the same distance vector to $S$. The complexity of Metric Dimension in graphs of bounded treewidth remained elusive in the past years. Recently, Bonnet and Purohit [IPEC 2019] showed that the problem is W[1]-hard under treewidth parameterization. In this work, we strengthen their lower bound to show that Metric Dimension is NP-hard in graphs of treewidth 24.