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 determinis...

Example 1 Undirected Hamiltonian Path (UHP) and Undirected Hamiltonian Circuit (UHC) are the same problems as the corresponding problems for directed graphs, but the new problems are for undirected graphs. Claim 2 Given a polynomial time algorithm for Undirected Hamiltonian Path (UHP) there is a polynomial time algorithm for Directed Hamiltonian Path. 1. A DHP builds the triple (H, p, q), where...

We study the question of whether the class DisjNP of disjoint pairs (A,B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provi...