Theory of Integer Programming

Suppose that we have a linear program max{cx : Ax ≤ b, x ≥ 0} (P) where A is an m × n matrix, c an n-dimensional row vector, b an m-dimensional column vector, and x an n-dimensional column vector of variables. Now, we add in the restriction that some of the variables in (P) must take integer values. If some but not all variables are integer, we have a Mixed Integer Program (MIP), written as max cx+ hy s.t. (MIP) Ax+Gy ≤ b x ≥ 0 y ≥ 0 and integer. where A is again m× n matrix, G is m× p, h is a p row vector, and y is a p column vector of integer variables. If all variables are integer, then we have an Integer Program (IP), written as max cx s.t. (IP) Ax ≤ b x ≥ 0 and integer.