Before revision:  Let us consider a different, seemingly unrelated problem in graph theory: given three arbitrary square matrices P, I and C with elements in \mathbb{R\cup\{-\infty\}, find all real values of parameter \lambda such that the parametric directed graph having arcs weights of the form w((j,i)) = \max(P_{ij} + \lambda, I_{ij} - \lambda, C_{ij}) (for all arcs (j,i)) }  does not contain circuits with positive weight.
After revision:  Let us consider a different, seemingly unrelated problem in graph theory: given three arbitrary square matrices P, I and C with elements in \mathbb{R\cup\{-\infty\}, find all real values of parameter \lambda such that the parametric directed graph having arcs weights of the form w((j,i)) = \max(P_{ij} + \lambda, I_{ij} - \lambda, C_{ij}) (for all arcs (j,i)) } In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP) consists in finding all the values of a parameter such that a particular parametric weighted directed graph does not contain circuits with positive weight.