Given a complete directed graph $G$ with weights on the vertices and on the arcs, a $\theta$-improper $k$-coloring of $G$ is an assignment of at most $k$ different colors to the vertices of $G$ such that the weight of every vertex $v$ is greater, by a given factor ${1}/{\theta}$, than the sum of the weights on the arcs $(u,v)$ entering $v$, with the tail $u$ of the same color as $v$. For a given real number $\theta$ and an integer $k$, the Partial Directed Weigthed Improper Coloring Problem ($(\theta,k)$-PDWICP) is to determine the order of the largest induced subgraph $G'$ of $G$ such that $G'$ admits a $\theta$-improper $k$-coloring. The $(\theta,k)$-PDWICP appears as a natural model when solving channel assignment problems that aim to maximize the number of simultaneously communicating mobiles in a wireless network. In this paper, we compare integer programming approaches to solve this $\mathcal{NP}$-hard problem to optimality. A polynomial-time computable upper bound inspired by the Lovász $\vartheta(G)$ function, and valid inequalities based on the knapsack and set-packing problems are embedded in a branch-and-bound procedure. Comparisons with a branch-and-price scheme are reported.