Highly parallel approximations for inherently sequential problems

In this work we study classes of optimization problems that require inherently sequential algorithms to solve exactly but permit highly parallel algorithms for approximation solutions. NC is the class of computational problems decidable by a logarithmic space uniform family of Boolean circuits of bounded fan-in, polynomial size, and polylogarithmic depth. Such problems are considered both “efficient” (since NC ⊆ P) and “highly parallel” (since we might consider each gate in the circuit to be a processor working in parallel and the small depth of the circuit a small number of steps). By contrast, problems which are P-complete (under logarithmic space or even NC many-one reductions) are considered “inherently sequential”. Furthermore, all NP-hard and PSPACE-hard problems are also inherently sequential, since P ⊆ NP ⊆ PSPACE. Just as we hope to find efficient approx mation algorithms for optimization problems for which it is intractable to compute an exact solution, so too do we hope to find efficient and highly parallel approximation algorithms for optimization problems for which computing the exact solution is inherently sequential. (However, just like hardness of approximation for NP-hard problems, in some cases even approximating a solution is inherently sequential!)