Reconfiguration in Curve Arrangements to Reduce Self-Intersections and Popular Faces

Florestan Brunck, Hsien-Chih Chang, Maarten Löffler, Tim Ophelders, Lena Schlipf

Abstract:

We study reconfiguration in curve arrangements, where a subset of the crossings are marked as {\em switches} which have three possible states, and the goal is to set the switches such that the resulting curve arrangement has few self-intersections, or few faces that are incident to the same curve multiple times (a.k.a.\ {\em popular faces}).
Our results are that these problems are NP-hard, but FPT in the number of switches. Minimizing self-intersections is also FPT in the number of {\em non-switchable} crossings; for minimizing popular faces this problem remains open.
Our results can be applied to generating {\em curved nonograms}, a type of logic puzzle that has received some attention lately. Specifically, our results make it possible to efficiently convert {\em expert} puzzles into {\em advanced} puzzles (or determine that this is impossible).