On the Complexity of Combinatorial and Metafinite Generating Functions of Graph Properties in the Computational Model of Blum, Shub and Smale

• Authors: Johann A. Makowsky;Klaus Meer
• Venue: Proceedings of the 14th Annual Conference of the EACSL on Computer Science Logic
• Year: 2000

We present a unified frame work for the study of the complexity of counting functions and multivariate polynomials such as the permanent and the hamiltonian in the computational model of Blum, Shub and Smale. For PIR we introduce complexity classes GenPIR and CGenPIR. The class GenPIR consists of the generating functions for graph properties (decidable in polynomial time) first studied in the context of Valiant's VNP by Bürgisser. CGenPIR is an extension of GenPIR where the graph properties may be subject to numeric constraints. We show that GenPIR ⊆ CGenPIR ⊆ EXPTIR and exhibit complete problems for each of these classes. In particular, for (n × n) matrices M over IR, ham(M) is complete for GenPIR, but the exact complexity of per(M) ∈ GenPIR remains open. Complete problems for CGenPIR are obtained by converting optimization problems which are hard to approximate, as studied by Zuckerman, into corresponding generating functions. Finally, we enlarge once more the class of generating functions by allowing additionally a kind of non-combinatorial counting. This results in a function class Met-GenPIR for which we also give a complete member: evaluating a polynomial in the zeros of another one and summing up the results. The class Met-GenPIR is also a generalization of #PIR, introduced by Meer, [Mee97]. Due to lackof space we will prove here only the Met-GenPIR result. In the full paper also the other theorems will be established rigorously.