Browse Theses

The hardest problems in the complexity class NP are called NP-complete. However, not all NP-complete problems are equally hard to solve from the average point of view. For example, the Hamiltonian circuit problem has been shown to be solvable deterministically in polynomial time on the average, whereas the bounded tiling problem still remains hard to solve even on the average. We therefore need a thorough analysis of the average behavior of algorithms.

In response to this need, L. Levin initiated in 1984 a theory of average-case NP-completeness. Levin's theory deals with average-case NP-complete problems using polynomial-time many-one reductions. The reducibility is a method by which we can classify the distributional NP problems.

In this thesis, we develop a more general theory of average-case complexity to determine the relative complexity of all natural average-case intractable problems. We investigate structure of reducibilities, including a bounded-error probabilistic truth-table reducibility. We introduce a variety of relativizations of fundamental average-case complexity classes of distributional decision problems. These relativizations are essential when we attempt to expand our notion of average polynomial-time computability to develop a hierarchy above average NP problems.

Average-case analyses are very sensitive to the choice of probability distributions. We have observed that if the input probability distribution decays exponentially with size, for instance, all NP-complete problems are solved "fast" on the average. This phenomenon does not reflect a significant feature of average-case analysis. This thesis includes a thorough analysis of structural properties of feasibly computable distributions and feasibly samplable distributions.

In addition, one may ask how we can extract the essential average behavior of algorithms independent of the choice of probability distributions. To answer this question, this thesis introduces the new notion of quintessential computability, which expands the boundary of worst-case feasible computability (such as polynomial-time computability), and asserts the invariance of average-case complexity of algorithms regardless of which feasibly computable distributions are chosen. This thesis examines the hardness of this real computability and its structural properties.