Completeness for parity problems

Abstract

In this talk we shall review recent work on holographic algorithms and circuits. This work can be interpreted as offering formulations of the question of whether computations within such complexity classes as NP, ⊕P, BQP, or #P, can be efficiently computed classically using linear algebra. The central part of the theory is the consideration of gadgets that map simple combinatorial constraints into gates, assemblies of which can be evaluated efficiently using linear algebra. The combinatorial constraints that appear most fruitful to investigate are the simplest ones that correspond to problems complete in these complexity classes. With this motivation we shall in this note consider the parity class ⊕P for which our understanding of complete problems is particularly limited. For example, among the numerous search problems for which the existence of solutions can be determined in P and the counting problem is known to be #P-complete, the #P-completeness proof does not generally translate to a ⊕P-completeness proof. We observe that in one case it does, and enumerate several natural problems for which the complexity of parity is currently unresolved. We go on to consider two examples of NP-complete problems for which ⊕P-completeness can be proved but is not immediate: Hamiltonian circuits for planar degree three graphs, and satisfiability of read-twice Boolean formulae. © Springer-Verlag Berlin Heidelberg 2005.