The approximability of problems complete for P

Abstract

We examine here the existence of approximations in NC of P-complete problems. We show that many P-complete problems (such as UNIT, PATH, circuit value etc.) cannot have an approximating solution in NC for any value of the absolute performance ratio R of the approximation, unless P=NC. On the other hand, we exhibit of a purely combinatorial problem (the High Connectivity subgraph problem) whose behaviour with respect to fast parallel approximations is of a threshold type. This dichotomy in the behaviour of approximations of P-complete problems is for the first time revealed and we show how the tools of log-space reductions can be used to make inferences about the best possible performance of approximations of problems that are hard to parallelize.