On optimal approximability results for computing the strong metric dimension

The strong metric dimension of a graph was first introduced by Sebö and Tannier (2004) as an alternative to the (weak) metric dimension of graphs previously introduced independently by Slater (1975) and by Harary and Melter (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∗(20.287n)-time exact computation algorithm, admits a O(1.2738k+nk)-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 (105−21−ε)-approximation algorithm assuming P≠NP, does not admit a O∗(2o(n))-time exact computation algorithm assuming the exponential time hypothesis is true, and does not admit a O∗(no(k))-time exact computation algorithm if the strong metric dimension is at most k assuming the exponential time hypothesis is true.