Completeness and Reduction in Algebraic Complexity Theory by Peter Bürgisser

By Peter Bürgisser

One of crucial and profitable theories in computational complicated­ ity is that of NP-completeness. This discrete thought relies at the Turing laptop version and achieves a category of discrete computational prob­ lems in keeping with their algorithmic trouble. Turing machines formalize al­ gorithms which function on finite strings of symbols over a finite alphabet. against this, in algebraic versions of computation, the elemental computational step is an mathematics operation (or comparability) of components of a set box, for in­ stance of genuine numbers. Hereby one assumes distinct mathematics. In 1989, Blum, Shub, and Smale [12] mixed current algebraic versions of computation with the idea that of uniformity and built a idea of NP-completeness over the reals (BSS-model). Their paper created a renewed curiosity within the box of algebraic complexity and initiated new examine instructions. the final word target of the BSS-model (and its destiny extensions) is to unite classical dis­ crete complexity concept with numerical research and therefore to supply a deeper beginning of clinical computation (cf. [11, 101]). Already ten years prior to the BSS-paper, Valiant [107, one hundred ten] had proposed an analogue of the speculation of NP-completeness in a wholly algebraic body­ paintings, in reference to his well-known hardness end result for the everlasting [108]. whereas the a part of his conception according to the Turing strategy (#P-completeness) is now common and recognized one of the theoretical laptop technological know-how com­ munity, his algebraic completeness end result for the permanents obtained less attention.

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Of Thm. g. that Pn(l) = 0 for all n. t. the index set I. By Thm. 10, Cor. 12 the family h is complete. We are going to show that h is a p-projection of the family (CFpJ of cycle format polynomials. We write N:= (n+2)4. , tN 2:: n + 2. 12, UHC tN is a projection of CF PN' By monotonicity, we conclude that h n = UHC n is a projection of CF PN' Assume now n E I. Define rand m by r := pN(m) := maXi PN(i). Note that 2 :s: m :s: tN :s: n + 1. From N = L;ipN(i) :s: rtN we conclude that r 2:: (n + 2)3, hence n + mn 2 :s: r.

Thus (CF Pn) is not p-computable if CE>l. ' D For statements related to this proposition, which rely on weaker hypotheses, we refer to Chap. 5. 4 Graph Factors In the sequel let F denote a connected graph. Let the graph property F A( F) describe the graphs all of whose connected components are isomorphic to F. A spanning subgraph of a graph G which has the property FA(F) will be called an F-factor of G. The corresponding generating functions will be called the F -factor polynomials. We remark that this notion contains certain cycle format polynomials as special cases.

A subset J ~ m can be identified with the vector e E {O, l}m characterized by ei = 0 iff i E J. 2). 3 Closure Properties The complexity class VNP is closed under various natural operations. We have the following result due to Valiant [110]. 19 Let (fn), (9n) bep-definable, say fn E k[X 1, ... , Xv(n)]' Then: (1) (Sum and product) (fn + 9n) and (fn . 9n) are p-definable. (2) (Substitution) (fn(91 , ... ,9v(n))) is p-definable. (3) (Coefficient) If hn E k[Xu(n)+l,"" Xv(n)] is the coefficient of some power product X~l ...

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