The document discusses NP-complete problems and algorithms to solve them in polynomial time using a decision box approach. It examines permutation problems like Hamiltonian path and cycle, subset problems including minimum vertex cover, minimum dominating set, and maximum clique. It also discusses using polynomial-time reductions to show problems are NP-complete by reducing them from known NP-complete problems like SAT in polynomial time. The decision box algorithms check solutions in polynomial time to find optimal solutions.