Upward and Orthogonal Planarity are W[1]-hard Parameterized by Treewidth

TL;DR

This paper proves that upward and rectilinear planarity testing are W[1]-hard when parameterized by treewidth, resolving open questions and indicating no significantly faster algorithms are likely.

Contribution

It establishes W[1]-hardness of upward and rectilinear planarity testing parameterized by treewidth, using a novel analysis of the All-or-Nothing Flow problem.

Findings

W[1]-hardness of planarity testing problems parameterized by treewidth.

W[1]-hardness of the All-or-Nothing Flow problem on planar graphs.

Existing algorithms cannot be improved to sub-exponential in treewidth time unless ETH fails.

Abstract

Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both NP-complete, but XP when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known $n^{O(tw)}$-time algorithms cannot be improved to run in time $n^{o(tw)}$ unless ETH fails.