Publications (13)
Covering and Partitioning Complex Objects with Small Pieces
Anders Aamand, Mikkel Abrahamsen, Reilly Browne +6
We study the problems of covering or partitioning a polygon (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contain…
Reconfiguration of unit squares and disks: PSPACE-hardness in simple settings
Mikkel Abrahamsen, Kevin Buchin, Maike Buchin +5
We study two well-known reconfiguration problems. Given a start and a target configuration of geometric objects in a polygon, we wonder whether we can move the objects from the sta…
Bounding a Polygon by a Minimum Number of Vertices
Mikkel Abrahamsen, Jack Stade, Shuyi Yan +1
Suppose that a polygon is given as an array containing the vertices in counterclockwise order. We analyze how many vertices (including the index of each of these vertices) we n…
Probabilistically checkable proofs for the Existential Theory of the Reals
Jack Stade
We prove a PCP theorem for the existential theory of the reals, showing that MAX-ETR-INV is -hard to approximate to within some constant factor. The existential…
The Point-Boundary Art Gallery Problem is -hard
Jack Stade
We resolve the complexity of the point-boundary variant of the art gallery problem, showing that it is -complete, meaning that it is equivalent under polynomial…
Topological Universality of the Art Gallery Problem
Jack Stade, Jamie Tucker-Foltz
We prove that any compact semi-algebraic set is homeomorphic to the solution space of some art gallery problem. Previous works have established similar universality theorems, but h…
Better Neural Network Expressivity: Subdividing the Simplex
Egor Bakaev, Florestan Brunck, Christoph Hertrich +2
This work studies the expressivity of ReLU neural networks with a focus on their depth. A sequence of previous works showed that hidden layers are suffi…
NP-membership for the boundary-boundary art-gallery problem
Jack Stade
The boundary-boundary art-gallery problem asks, given a polygon representing an art-gallery, for a minimal set of guards that can see the entire boundary of (the wall of th…
Devil's Games and : Continuous Games complete for the First-Order Theory of the Reals
Lucas Meijer, Arnaud de Mesmay, Tillmann Miltzow +2
We introduce the complexity class Quantified Reals (). Let FOTR be the set of true sentences in the first-order theory of the reals. A language is in $\text…
Humanity's Last Exam
Long Phan, Alice Gatti, Ziwen Han +1144
Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achi…
Hardness of Packing, Covering and Partitioning Simple Polygons with Unit Squares
Mikkel Abrahamsen, Jack Stade
We show that packing axis-aligned unit squares into a simple polygon is NP-hard, even when is an orthogonal and orthogonally convex polygon with half-integer coordinates. I…
Shallower ReLU Network Representations via Exact Linear Algebra
Kilian RueÃ, Gennadiy Averkov, Florestan Brunck +7
We prove that the maximum of real numbers is exactly representable by a ReLU network with two hidden layers for every . The constructions are obtained by reducing the…
Two Tiling is Undecidable
Jack Stade
We show that the following problem is undecidable: given two polygonal prototiles, determine whether the plane can be tiled with rotated and translated copies of them. This improve…