Sparsification Lower Bound for Linear Spanners in Directed Graphs

TL;DR

This paper proves that for directed graphs, linear spanners cannot be significantly compressed in size unless unlikely complexity class collapses occur, establishing fundamental sparsification lower bounds.

Contribution

It establishes polynomial compression lower bounds for directed linear, additive, and multiplicative spanners, extending prior work to directed graphs and general functions.

Findings

No polynomial compression of size O(n^{2 - ε}) unless NP ⊆ coNP/poly

Lower bounds hold even for DAGs and arbitrary computable functions

Results extend to additive and multiplicative spanners in directed graphs

Abstract

For $\alpha \ge 1$, $\beta \ge 0$, and a graph $G$, a spanning subgraph $H$ of $G$ is said to be an $(\alpha, \beta)$-spanner if $\dist(u, v, H) \le \alpha \cdot \dist(u, v, G) + \beta$ 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 $\calO(n^{2 - \epsilon})$ for any $\epsilon > 0$ 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 $\alpha, \beta$ are \emph{any} computable functions of the distance being approximated.