  In this work, we study the computational complexity of massive gravity theory
via the `Complexity = Action' conjecture. Our system contains a particle moving
on the boundary of the black hole spacetime. It is dual to inserting a
fundamental string in the bulk background. Then this string would contribute a
Nambu-Goto term, such that the total action is composed of the Einstein-Hilbert
term, Nambu-Goto term and the boundary term. We shall investigate the time
development of this system, and mainly discuss the features of the Nambu-Goto
term affected by the graviton mass and the horizon curvature in different
dimensions. Our study could contribute interesting properties of complexity.
