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.

This research was done during the visit of the first author to Patras University, and is supported partially by a Spanish Research Scholarship and by the Ministry of Industry, Energy and Technology of Greece.