On optimal approximability results for computing the strong metric dimension

Abstract

The strong metric dimension of a graph was first introduced by Sebö and Tannier (Mathematics of Operations Research, 29(2), 383-393, 2004) as an alternative to the (weak) metric dimension of graphs previously introduced independently by Slater (Proc. 6 th Southeastern Conference on Combinatorics, Graph Theory, and Computing, 549-559, 1975) and by Harary and Melter (Ars Combinatoria, 2, 191-195, 1976), and has since been investigated in several research papers. However, the exact worst-case computational complexity of computing the strong metric dimension has remained open beyond being NP-complete. In this communication, we show that the problem of computing the strong metric dimension of a graph of n nodes admits a polynomial-time 2-approximation, admits a O * 2 0.287 n -time exact computation algorithm, admits a O 1.2738 k + n k -time exact computation algorithm if the strong metric dimension is at most k, does not admit a polynomial time (2 − ε)-approximation algorithm assuming the unique games conjecture is true, does not admit a polynomial time (10 √ 5 − 21 − ε)-approximation algorithm assuming P NP, does not admit a O * 2 o(n) -time exact computation algorithm assuming the exponential time hypothesis is true, and does not admit a O * n o(k) -time exact computation algorithm if the strong metric dimension is at most k assuming the exponential time hypothesis is true.

Strong resolving graphs: The realization and the characterization problems

“…Besides the theoretical results related to the strong metric dimension, a mathematical programming model [16] and metaheuristic approaches [17,28] for finding this parameter have been developed. Some complexity and approximation results are also known from the works [31] and [4], respectively. On the other hand, a fractional version of the strong metric dimension has been studied in [12,13,14].…”

On analyzing and evaluating privacy measures for social networks under active attack

“…In other words, this means that a social network K n guarantees that a user cannot be re-identified (based on the metric representation) with a probability higher than 1/(n − ) by an adversary controlling at most attacker nodes. For other related concepts for metric dimension of graphs, the reader may consult references such as [14,25,30].…”

Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results

“…Despite the fact that Theorem 2 directly gives an approximation result for the strong metric dimension, some extra work on approximating such parameter was published in [46]. There was proved, among other results, that the problem of computing the strong metric dimension of a graph of order n admits a O(2 0.287n )-time exact computation algorithm.…”