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- W865480013 abstract "One of the most important topics in structural complexity theory is the question, whether nondeterministic polynomial time for Turing machine computations is more powerful than deterministic polynomial time. Some insight into Cook’s hypothesis “P ≠ NP” is provided by polynomial time reductions which allow to single out the NP-complete problems as the hardest problems in NP, see Cook [124], Karp [297], and Levin [328]. This chapter discusses Valiant’s nonuniform algebraic analogue of NP-completeness [526, 529] which grew out of his studies of enumeration problems culminating in the theory of #P-completeness. The first section introduces Valiant’s algebraic complexity classes VP and VNP, as well as p-projections as an algebraic analogue of polynomial time reductions. The elements of these classes are certain families of multivariate polynomials over some fixed field k. Valiant’s hypothesis “VP ≠ VNP” gains interest by his theorem stating that the family PER = (PER n) of generic permanents is VNP-complete over any field of characteristic different from 2. The proof of this fundamental result, which is the main goal of this chapter, is based on an alternative characterization of VNP in terms of the expression or formula size of polynomials, see Sect. 21.2. In the third section it is shown that every polynomial of expression size u is a projection of PER2u+2. Sect. 21.4 presents a simplified proof of Valiant’s theorem based on a generalized Laplace expansion theorem. In the last section we prove Brent’s theorem [72] which describes the close connection between depth and expression size, as well as results of Hyafil [260] and Valiant, Skyum, Berkowitz, and Rackoff [530] estimating the depth of a polynomial in terms of its degree and complexity. On the basis of these results the extended Valiant hypothesis can be formulated in purely algebraic terms: for any fixed positive constant c there is no possibility of writing all generic permanents as PER n = DET m(n) (A), where A is an m(n) x m(n) matrix over k ⋃ Xμυ|μ,υ ϵ n and m(n) = 2O(logcn).KeywordsTuring MachineHamiltonian CycleCounting ProblemMultivariate PolynomialPolynomial Time ReductionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
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- W865480013 date "1997-01-01" @default.
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- W865480013 title "P Versus NP: A Nonuniform Algebraic Analogue" @default.
- W865480013 doi "https://doi.org/10.1007/978-3-662-03338-8_21" @default.
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