On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs

Authors Florian Barbero, Lucas Isenmann, Jocelyn Thiebaut

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Florian Barbero
  • LIRMM, Université de Montpellier, 161 rue Ada, 34095, Montpellier, France
Lucas Isenmann
  • LIRMM, Université de Montpellier, 161 rue Ada, 34095, Montpellier, France
Jocelyn Thiebaut
  • LIRMM, Université de Montpellier, 161 rue Ada, 34095, Montpellier, France

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Florian Barbero, Lucas Isenmann, and Jocelyn Thiebaut. On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations r-IC and r-LD where the closed neighborhood is considered up to distance r. It also contains Metric Dimension (MD) and its refinement r-MD in which the distance between two vertices is considered as infinite if the real distance exceeds r. Note that while IC = 1-IC and LD = 1-LD, we have MD = infty-MD; we say that MD is not local. In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We focus on two families of problem from the meta-problem: the first one, called bipartite gifted local, contains r-IC, r-LD and r-MD for each positive integer r while the second one, called 1-layered, contains LD, MD and r-MD for each positive integer r. We have: - the 1-layered problems are NP-hard even in bipartite apex graphs, - the bipartite gifted local problems are NP-hard even in bipartite planar graphs, - assuming ETH, all these problems cannot be solved in 2^{o(sqrt{n})} when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in 2^{o(n)} on bipartite graphs, - even restricted to bipartite graphs, they do not admit parameterized algorithms in 2^{O(k)} * n^{O(1)} except if W[0] = W[2]. Here k is the solution size of a relevant identifying set. In particular, Metric Dimension cannot be solved in 2^{o(n)} under ETH, answering a question of Hartung in [Sepp Hartung and André Nichterlein, 2013].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Problems, reductions and completeness
  • identifying code
  • resolving set
  • metric dimension
  • distance identifying set
  • parameterized complexity
  • W-hierarchy
  • meta-problem
  • hitting set


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