Metric Planarity Testing is Hard

Abstract

A key milestone in algorithmic graph theory is Hopcroft and Tarjan's 1974 linear-time algorithm for graph planarity testing. At a workshop in 2002, Linial and Rabinovich asked for either a polynomial-time algorithm or an NP-hardness proof for metric planarity testing: given a metric space $(X,d_X)$, determine whether there exists an edge-weighted planar graph $G=(V,E,w)$ with $X\subseteq V$ that realizes the distances in $X$, meaning that $d_G(x,y)=d_X(x,y)$ for every $x,y\in X$. This question arose in the context of research on metric embeddings and their applications in computer science since the mid-1990s. Over the past few decades, polynomial-time algorithms have been developed for several special cases. In this work, we show that metric planarity testing is NP-hard, in sharp contrast to graph planarity testing. Our reduction yields a stronger result: under the Exponential Time Hypothesis (ETH), no $2^{o(n)}$-time algorithm for metric planarity testing exists, where $n$ is the number of points in the metric. To complement our ETH lower bound, we give two algorithmic results: (i) an algorithm for computing a metric planar embedding in $2^{\tilde{O}(n^2)}$ time and (ii) a $(1+ε)$-approximation algorithm in $2^{\tilde{O}(n)}$ time for any fixed $ε\in(0,1)$. The second algorithm produces a planar graph that preserves the input metric up to a factor of $(1+ε)$.