Improved lowness results for solvable black-box group problems

Abstract

In order to study the complexity of computational problems that arise from group theory in a general framework, Babai and Szemerédi [4, 2] introduced the theory of black-box groups. They proved that several problems over black-box groups are in the class NP ∩ co-AM, thereby implying that these problems are low (powerless as oracle) for ∑ ^p₂ and hence cannot be complete for NP unless the polynomial hierarchy collapses.

In [1], Arvind and Vinodchandran study the counting complexity of a number of computational problems over solvable groups. Using a constructive version of a fundamental structure theorem about finite abelian groups and a randomized algorithm from [3] for computing generator sets for the commutator series of any solvable group, they prove that these problems are in randomized versions of low complexity counting classes like SPP and LWPP and hence low for the class PP.

In this paper, we improve the upper bounds of [1] for these problems. More precisely, we avoid the randomized algorithm from [3] for computing the commutator series. This immediately places all these problems in either SPP or LWPP. These upper bounds imply lowness of these problems for classes other than PP. In particular, SPP is low for all gap-definable counting classes [9] (PP, C=P, Mod_kP etc) and LWPP is known to be low for PP and C=P. These results are in favor of the belief that these problems are unlikely to be complete for NP.