This work studies the expressivity of ReLU neural networks with a focus on their depth. A sequence of previous works showed that ⌈ log2(n+1) ⌉ hidden layers are sufficient to compute all continuous piecewise linear (CPWL) functions on ℝn. Hertrich, Basu, Di Summa, and Skutella (NeurIPS ’21 / SIDMA ’23) conjectured that this result is optimal in the sense that there are CPWL functions on ℝn, like the maximum function, that require this depth. We disprove the conjecture and show that ⌈log3(n−1)⌉+1 hidden layers are sufficient to compute all CPWL functions on ℝn.

We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.

We study reconfiguration in curve arrangements, where a subset of the crossings are marked as 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. 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 non-switchable crossings; for minimizing popular faces this problem remains open. Our results can be applied to generating curved nonograms, a type of logic puzzle that has received some attention lately. Specifically, our results make it possible to efficiently convert expert puzzles into advanced puzzles (or determine that this is impossible).

Abstract Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform $$\delta >0$$ δ > 0 such that, at any step of the subdivision, all the triangle angles lie in the interval $$(\delta ,\pi -\delta )$$ ( δ , π - δ ) . Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.