The author proposes an innovative approach that utilizes a single parameter as the gap between the objective values of a linear program and its associated linear integer, binary, or mixed programs (LIP, LBP, or/and LMP) with the concept of Generalized Gaussian Elimination (GGE) to resolve the feasibility of the associated linear Integer, binary, or mixed programs as to obtain the desired optimal solution if such a solution for LIP, LBP, or LMP does exist. Such an innovative LIP, LBP, or LMP solution technique does not require the traditional branch and bound (BandB) technique and it offers a computational complexity that is comparable to that of the GGE solution technique itself. Note that the computational complexity of the GGE approach is comparable to that of the original Gaussian Elimination (GE) for system of linear equalities. Sample LIP and LBP using this parameterized GGE to find their optimal solutions that match exactly to the answers obtained by the traditional BandB technique are provided to illustrate the correctness and simplicity of such a parameterized GGE (PGGE) approach for solving LIP, LBP, or LMP. Consequently, this PGGE is a new and effective solution technique much more powerful than the traditional BandB technique for LIP, LBP or LMP. Applying such a parameterized GGE solution technique to problems in the NP-Complete (NPC) group, one may be able to determine the overall computational complexity of the NP class and provide insight as to whether or not NP is also P? Furthermore, such a parameterized GGE technique is also applicable to resolve the feasibility of integer, binary, or mixed differential variation inequalities.
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