The NP -hardness of the satisfiability problem was demonstrated by exhibiting the existence of a polynomial time reduction, from each problem in NP to the satisfiability problem. A proof by reduction is possible because the composition of polynomial time reductions is also a polynomial time reduction. The proofs of the following two theorems exhibit the NP -hardness of the problems in question by means of reduction.
M can be of polynomial time complexity.
Therefore the 0 - 1 knapsack problem is in NP . E a system S of linear equations of the following form can be constructed. The system S can be represented in a vector form as follows. E is an instance of the 3-satisfiability problem. The previous examples of NP -complete problems deal with Boolean expressions and linear equations.