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

Abstract

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 problems from the meta-problem: the first one, called 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: (1) the 1-layered problems are NP-hard even in bipartite apex graphs, (2) the local problems are NP-hard even in bipartite planar graphs, (3) 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, and (4) except if $${\mathsf{W}[0]} = {\mathsf{W}[2]}$$ , they do not admit parameterized algorithms in $$2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}$$ even when restricted to bipartite graphs. 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 and Nichterlein (Proceedings of the 28th conference on computational complexity, CCC, 2013).