关于VNP-完全性和边界复杂性的注记
A note on VNP-completeness and border complexity
摘要：
在1979年Valiant介绍了复杂性类VNP的p-可定义的家庭的多项式，他定义了减少的概念被称为p-投影，他证明了永久多项式和哈密尔顿循环多项式是VNP完全下p-投影。2001年，Mulmuley和Sohoni（以及独立的Bürgisser）将边界复杂性的概念引入到多项式的代数复杂性的研究中。在这个代数机器模型中，允许近似，而不是坚持精确计算。这给了VNP拓扑空间的结构。本文研究VNP-完全多项式的集合VNPC。我们发现，补充VNP \ VNPC位于密集的VNP。令人惊讶的是，我们还证明了VNPC在VNP中是稠密的。我们证明了复杂性类VF、VBP和VP的类似陈述。VNP \ VNPC的密度适用于几种不同的约简概念：p-投影、边界p-投影、c-约简和边界c-约简。我们比较了这些减少下的完整性概念的关系，并分离出大多数相应的集合。Bringmann，Ikenmeyer和Zuiddam（JACM 2018）引入了边界缩小概念。我们的论文是第一个结构化的研究边界减少的概念。
英文摘要：
In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, he defined the reduction notion known as p-projection and he proved that the permanent polynomial and the Hamiltonian cycle polynomial are VNP-complete under p-projections. In 2001 Mulmuley and Sohoni (and independently Bürgisser) introduced the notion of border complexity to the study of the algebraic complexity of polynomials. In this algebraic machine model, instead of insisting on exact computation, approximations are allowed. This gives VNP the structure of a topological space. In this short note we study the set VNPC of VNP-complete polynomials. We show that the complement VNP \ VNPC lies dense in VNP. Quite surprisingly, we also prove that VNPC lies dense in VNP. We prove analogous statements for the complexity classes VF, VBP, and VP. The density of VNP \ VNPC holds for several different reduction notions: p-projections, border p-projections, c-reductions, and border c-reductions. We compare the relationship of the completeness notions under these reductions and separate most of the corresponding sets. Border reduction notions were introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Our paper is the first structured study of border reduction notions.
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