Parallelism with limited nondeterminism

Since parallel computing is again becoming a topic of interest in computer science, it is important to revisit the theoretical foundations of highly parallel computing. “Inherently sequential” computational problems see no significant speedup when run on highly parallel computers. Just as there are efficient approximations for intractable optimization problems, so too are there efficient and highly parallel approximations for optimization problems that are tractable but inherently sequential. (From here forward, we write “parallel” instead of “efficient and highly parallel” and “sequential” instead of “inherently sequential” . ) For example, the problem of computing the optimal vector in a positive linear program, a problem relevant to distributed flow control within a network of routers, is sequential, but a vector very close to the optimal one can be computed quickly in parallel. We intend to develop the theory of structural complexity for parallel approximations for tractable sequential problems. This area has not been well-studied, and when it has been studied, the results focus mostly on parallel approximations for intractable optimization problems (that is, NC approximations for NP-complete problems), not parallel approximations for tractable sequential problems (that is, NC approximations for P-complete problems). This prospectus describes work we have already completed and work that remains in developing this theory. The two sections below discuss two main approaches to proving the limitations of parallel approximations for sequential problems. The first section discusses the complexity classes associated with parallel approximation algorithms, the