Introduction to Np-completeness

We use time-bounded (deterministic and nondeterministic) Turing machines to study computational complexity of decision problems. In this model, a Turing machine has two different halting states, called the accepting state and the rejecting state, respectively. We say that a Turing machine M is bounded by time T if for any input x, M on x always halts in T (|x|) moves (steps). If M is deterministic and M on input x halts at the accepting state, then we say that x is accepted by M . If M is nondeterministic and there exists at least one computation path of M on input x that halts on the accepting state, then we say that x is accepted by M . We are particularly interested in polynomial-time bounds for the following reasons.