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A quadratic lower bound for homogeneous algebraic branching programs
computational complexity ( IF 1.4 ) Pub Date : 2019-06-08 , DOI: 10.1007/s00037-019-00186-3
Mrinal Kumar

AbstractAn algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in $$\mathbb{F}[x_1, x_2, \ldots , x_n]$$F[x1,x2,…,xn]. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial $$x^n_1 + x^n_2 + \cdots + x^n_n$$x1n+x2n+⋯+xnn has at least $$\Omega(n^2)$$Ω(n2) vertices (and hence edges). To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of $$\Omega(n \,{\rm log}\, n)$$Ω(nlogn) on the number of edges in a general (possibly non-homogeneous) ABP, which follows from the classical results of Strassen (Numer Math 20:238–251, 1973) and Baur and Strassen (Theor Comput Sci 22:317–330, 1983). On the way, we also get an alternate and unified proof of an $$\Omega(n \,{\rm log}\, n)$$Ω(nlogn) lower bound on the size of a homogeneous arithmetic circuit (follows from the work of Strassen (1973) and Baur & Strassen (1983)), and an n/2 lower bound $$(n \,{\rm over}\, \mathbb{R})$$(noverR) on the determinantal complexity of an explicit polynomial (Mignon and Ressayre in Int Math Res Notes 2004(79):4241–4253, 2004; Cai et al. in Comput Complex 19(1):37–56, 2010, http://dx.doi.org/10.1007/s00037-009-0284-2; Yabe in CoRR, 2015, http://arxiv.org/abs/1504.00151). These are currently the best lower bounds known for these problems for any explicit polynomial and were originally proved nearly two decades apart using seemingly different proof techniques.

中文翻译:

齐次代数分支程序的二次下界

摘要代数分支程序 (ABP) 是一个有向无环图,具有起始顶点 s 和结束顶点 t,每条边的权重为 $$\mathbb{F}[x_1, x_2, \ldots , x_n]$$F[x1,x2,…,xn]。ABP 以自然的方式计算多项式,作为从 s 到 t 的所有路径的权重之和,其中路径的权重是路径中边权重的乘积。如果在每个顶点计算的多项式是齐次的,则称 ABP 是齐次的。在本文中,我们证明了任何计算多项式 $$x^n_1 + x^n_2 + \cdots + x^n_n$$x1n+x2n+⋯+xnn 的齐次代数分支程序至少有 $$\Omega(n^ 2)$$Ω(n2) 个顶点(以及边)。据我们所知,这似乎是一般齐次 ABP 顶点数量的第一个非平凡超线性下界,并略微改善了已知的下界 $$\Omega(n \,{\rm log}\, n)$ $Ω(nlogn) 在一般(可能是非同质的)ABP 中的边数,这是根据 Strassen (Numer Math 20:238–251, 1973) 和 Baur and Strassen (Theor Comput Sci 22: 317–330, 1983)。在此过程中,我们还得到了一个关于同构算术电路大小的 $$\Omega(n \,{\rm log}\, n)$$Ω(nlogn) 下界的替代和统一证明(来自Strassen (1973) 和 Baur & Strassen (1983) 的工作,以及 n/2 下界 $$(n \,{\rm over}\, \mathbb{R})$$(noverR) 在行列式上显式多项式的复杂性(Mignon 和 Ressayre Int Math Res Notes 2004(79):4241–4253, 2004; Cai et al. 在 Comput Complex 19(1):37–56, 2010, http://dx.doi.org/10.1007/s00037-009-0284-2; Yabe 在 CoRR,2015 年,http://arxiv.org/abs/1504.00151)。这些是目前已知的对于任何显式多项式的这些问题的最佳下界,并且最初使用看似不同的证明技术相隔近二十年被证明。
更新日期:2019-06-08
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