  The paper considers the problem of a leader that seeks to optimally influence
the opinions of agents in a directed network through connecting with a limited
number of the agents ("direct followers"), possibly in the presence of a fixed
competing leader. The settings involving a single leader and two competing
leaders are unified into a general combinatoric optimization problem, for which
two heuristic approaches are developed. The first approach is based on a convex
relaxation scheme, possibly in combination with the $\ell_1$-norm
regularization technique, and the second is based on a greedy selection
strategy. The main technical novelties of this work are in the establishment of
supermodularity of the objective function and convexity of its continuous
relaxation. The greedy approach is guaranteed to have a lower bound on the
approximation ratio sharper than $(1-1/e)$, while the convex approach can
benefit from efficient (customized) numerical solvers to have practically
comparable solutions possibly with faster computation times. The two approaches
can be combined to provide improved results. In numerical examples, the
approximation ratio can be made to reach $90\%$ or higher depending on the
number of direct followers.
