On the Complexity of Combinatorial andMeta nite Generating
نویسنده
چکیده
We present a uniied 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 PI R we introduce complexity classes GenPI R and CGenPI R: The class GenPI R consists of the generating functions for graph properties (decidable in polynomial time) rst studied in the context of Valiant's VNP by B urgisser. CGenPI R is an extension of GenPI R where the graph properties may be subject to numeric constraints. We show that GenPI R CGenPI R EXPTI R and exhibit complete problems for each of these classes. In particular, for (n n) matrices M over IR, ham(M) is complete for GenPI R, but the exact complexity of per(M) 2 GenPI R remains open. Complete problems for CGenPI 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-GenPI 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-GenPI R is also a generalization of ]PI R, introduced by Meer, Mee97]. Due to lack of space we will prove here only the Met-GenPI R result. In the full paper also the other theorems will be established rigorously.
منابع مشابه
Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup
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تاریخ انتشار 2000