On the Distance Identifying Set meta-problem and applications to the complexity of identifying problems on graphs

TL;DR

This paper introduces the Distance Identifying Set meta-problem, unifying various graph vertex identification problems, and establishes their computational complexity bounds, including NP-hardness and ETH-based lower bounds.

Contribution

It formalizes the Distance Identifying Set meta-problem, provides generic reductions from Hitting Set, and proves complexity bounds for multiple related problems, including Metric Dimension.

Findings

NP-hardness in bipartite apex and planar graphs

ETH-based lower bounds on solution time

Metric Dimension cannot be solved in subexponential time under ETH

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 mainly 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 2013.