Chapter

Algorithms – ESA 2012

Volume 7501 of the series Lecture Notes in Computer Science pp 419-430

On the Complexity of Metric Dimension

  • Josep DíazAffiliated withDepartament de Llenguatges i Sistemes Informatics, UPC
  • , Olli PottonenAffiliated withDepartament de Llenguatges i Sistemes Informatics, UPC
  • , Maria SernaAffiliated withDepartament de Llenguatges i Sistemes Informatics, UPC
  • , Erik Jan van LeeuwenAffiliated withDept. Computer, Control, Managm. Eng., Sapienza University of Rome

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Abstract

The metric dimension of a graph G is the size of a smallest subset L ⊆ V(G) such that for any x,y ∈ V(G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomial-time solvable on trees, and to have a logn-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.