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

Abstract

We present a unified framework 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 P _ℝ we introduce complexity classes Gen P _ℝ and CGen P _ℝ. The class Gen P _ℝ consists of the generating functions for graph properties (decidable in polynomial time) first studied in the context of Valiant’s VNP by Bürgisser. CGen P _ℝ is an extension of Gen P _ℝ where the graph properties may be subject to numeric constraints.

We show that Gen P _ℝ ⊆ CGen P _ℝ ⊆ EXPT _ℝ and exhibit complete problems for each of these classes. In particular, for (n × n) matrices M over ℝ, ham(M) is complete for Gen P _ℝ, but the exact complexity of per(M) ∈ Gen P _ℝ remains open. Complete problems for CGen P _ℝ 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-Gen P _ℝ 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-Gen P _ℝ is also a generalization of #P _ℝ, introduced by Meer, [Mee97].

Due to lack of space we will prove here only the Met-Gen P _ℝ result. In the full paper also the other theorems will be established rigorously.

Partially supported by by the Fund for Promotion of Research of the Technion-Israeli Institute of Technology