Optimizing k-Planarity (Graph Drawing Contest Abstract)

Abstract

An embedding of a graph is k-planar if every edge has at most k crossings. The 2026 Graph Drawing Challenge gave a one-hour time window to create minimal k-planar straight-line embeddings for 15 graphs.
I present the winning approach. Initial layouts come from several construction heuristics and are then optimized by a tuned local search or by simulated annealing. Both heuristics move one node at a time, so the rate at which candidate moves can be evaluated is crucial for performance. A uniform lookup grid restricts the computation of each node movement to the edges it can affect, and the evaluation aborts as soon as the move is known to fail. Together these two optimizations reach hundreds of thousands of proposals per second on all competition instances. That was enough to run every instance more than 20 times within the time limit, which let me take or tie the best score on all 15 graphs.