On Quasilinear-Time Complexity Theory

This paper furthers the study of quasi-linear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilinear-time hierarchy. Whereas all previously known versions of the Valiant-Vazirani reduction from NP to parity run in quadratic time, we give a new construction using error-correcting codes that runs in quasilinear time. We show, however, that the important equivalence between search problems and decision problems in polynomial time is unlikely to carry over: if search reduces to decision for SAT in quasi-linear time, then all of NP is contained in quasi-polynomial time. Other connections are made to work by Stearns and Hunt on “power indices” of NP languages, and to work on bounded-query Turing reductions and helping by robust oracle machines.