Resource Bounds and Subproblem Independence

There are several reasons why some NP-hard problems can be solved deterministically in 2^{O(n^a)} time for some a, 0 < a < 1. One reason is that the problem can be solved with a nondeterministic procedure which uses only O(n^a) space. A second reason is that each instance of the problem exhibits a high degree of "subproblem independence" (specifically O(n^a) line channelwidth or treewidth). A third is that each instance of the problem can be solved using nondeterministic straight-line programs where the number of variables is only O(n^a). In this paper we show that, after imposing q-linear bounds on the computation time and obliviousness constraints on the nondeterminism, these three reasons are essentially equivalent. (q-Linear functions are similar to functions of the form n (log n)^k.) Specifically, the problem classes associated with these reasons are interchangeable using reductions which are simultaneously polynomial time and q-linear size-bounded. We call these PQ-reductions. The classes of problems solvable by these methods we call NQn^a. Because these classes are defined in a very general way, they include a rich variety of problems. We take this as evidence that these classes have "power index" a (i.e., that 2^{O(n^a)} is also a lower time bound). On the other hand, the easiness of all these problems can be derived from "subproblem independence" considerations, and thus the techniques associated with subproblem independence are seen to be more widely applicable than one might expect.