EXPTIME

Abstract: The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P. >

Abstract: We look at several problems from areas such as network flows, game theory, artificial intelligence, graph theory, integer programming and nonlinear programming and show that they are related in that any one of these problems is solvable in polynomial time if all the others are, too. At present, no polynomial time algorithm for these problems is known. These problems extend the equivalence class of problems known as P-Complete. The problem of deciding whether the class of languages accepted by polynomial time nondeterministic Turing machines is the same as that accepted by polynomial time deterministic Turing machines is related to P-Complete problems in that these two classes of languages are the same if each P-Complete problem has a polynomial deterministic solution. In view of this, it appears very likely that this equivalence class defines a class of problems that cannot be solved in deterministic polynomial time.

Abstract: This paper investigates the computational complexity of binary sequences as measured by the rapidity of their generation by multitape Turing machines. A "translational" method which escapes some of the limitations of earlier approaches leads to a refinement of the established hierarchy. The previous complexity classes are shown to possess certain translational properties. An related hierarchy of complexity classes of monotonic functions is examined

Abstract: @ PTAPE. We show that all NP complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. Thus showing that, inspite of the radically different origins and attempted simplification of these sets, all the known NP complete sets are identical but for polynomially time bounded permutations.Furthermore, if all NP complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be NP complete, nor can any sparse language over an arbitrary alphabet be NP complete. We show that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet. Similarly, we show that the hardest context-sensitive languages cannot be sparse. We also relate the existence of sparse complete sets to the existence of simple combinatorial circuits for the corresponding truncated recognition problem of these languages.