Sparsification Lower Bound for Linear Spanners in Directed Graphs

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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