We consider a routing problem which plays an important role in several applications, primarily in communication network planning and VLSI layout design. The original underlying graph algorithmic task is called Disjoint Paths problem. In most applications, one can encounter its capacitated generalization, which is known as the Unsplitting Flow problem. These algorithmic tasks are very hard in general, but various efficient (polynomial-time) approximate solutions are known. Nevertheless, the approximations tend to be rather complicated, often rendering them impractical in large, complex networks. Our goal is to present a solution that provides a simple, efficient algorithm for the unsplittable flow problem in large directed graphs. The simplicity is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm of approximation: rather than giving up finding an exact solution, we restrict the solution space to its most important subset and exclude those that are marginal in some sense. Then we find the exact optimum efficiently within the subset. Specifically, the sacrificed parts (i.e., the marginal instances) only contain scenarios where some edges are very close to saturation. Therefore, the excluded part is not significant, since the excluded almost saturated solutions are typically undesired in practical applications, anyway.