有向图中线性生成子的稀疏化下界
Sparsification Lower Bound for Linear Spanners in Directed Graphs
摘要：
对于$α\ge 1$、$β\ge 0$和图$G$，如果$\dist(u, v, H) \le α\cdot \dist(u, v, G) + β$对任意一对顶点$u$和$v$成立，则称$G$的生成子图$H$是$(α, β)$ -图。这些类型的spectrum，称为\emph{linear spanners}，概括了\emph{additive spanners}和\emph{multiplicative spanners}。最近，Fomin，Golovach，Lochet，Misra，Saurabh和Sharma发起了有向图的加法和乘法空间的研究（IPEC $2020$）。在这篇文章中，我们继续这条线的研究和证明，\textsc{Directed Linear Spanner}参数化的顶点$n$的数量不承认多项式压缩的大小$\calO(n^{2 - ε})$的任何$ε> 0$，除非$\NP \subseteq \coNP/poly$。我们发现，类似的结果举行\textsc{Directed Additive Spanner}和\textsc{Directed Multiplicative Spanner}问题。即使当输入是有向无环图并且$α, β$是被近似的距离的\emph{any}可计算函数时，该稀疏化下限也保持。
英文摘要：
For $α\ge 1$, $β\ge 0$, and a graph $G$, a spanning subgraph $H$ of $G$ is said to be an $(α, β)$-spanner if $\dist(u, v, H) \le α\cdot \dist(u, v, G) + β$ holds for any pair of vertices $u$ and $v$. These type of spanners, called \emph{linear spanners}, generalizes \emph{additive spanners} and \emph{multiplicative spanners}. Recently, Fomin, Golovach, Lochet, Misra, Saurabh, and Sharma initiated the study of additive and multiplicative spanners for directed graphs (IPEC $2020$). In this article, we continue this line of research and prove that \textsc{Directed Linear Spanner} parameterized by the number of vertices $n$ admits no polynomial compression of size $\calO(n^{2 - ε})$ for any $ε> 0$ unless $\NP \subseteq \coNP/poly$. We show that similar results hold for \textsc{Directed Additive Spanner} and \textsc{Directed Multiplicative Spanner} problems. This sparsification lower bound holds even when the input is a directed acyclic graph and $α, β$ are \emph{any} computable functions of the distance being approximated.
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