Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry: Theory and Applications. Major revision with two new authors, solving the open problem about triangular face

Towards Generic Scalable Parallel Combinatorial Search

Combinatorial search problems in mathematics, e.g. in finite geometry, are notoriously hard; a state-of-the-art backtracking search algorithm can easily take months to solve a single problem. There is clearly demand for parallel combinatorial search algorithms scaling to hundreds of cores and beyond. However, backtracking combinatorial searches are challenging to parallelise due to their sensitivity to search order and due to the their irregularly shaped search trees. Moreover, scaling parallel search to hundreds of cores generally requires highly specialist parallel programming expertise. This paper proposes a generic scalable framework for solving hard combinatorial problems. Key elements are distributed memory task parallelism (to achieve scale), work stealing (to cope with irregularity), and generic algorithmic skeletons for combinatorial search (to reduce the parallelism expertise required). We outline two implementations: a mature Haskell Tree Search Library (HTSL) based around algorithmic skeletons and a prototype C++ Tree Search Library (CTSL) that uses hand coded applications. Experiments on maximum clique problems and on a problem in finite geometry, the search for spreads in H(4,2^2), show that (1) CTSL consistently outperforms HTSL on sequential runs, and (2) both libraries scale to 200 cores, e.g. speeding up spreads search by a factor of 81 (HTSL) and 60 (CTSL), respectively. This demonstrates the potential of our generic framework for scaling parallel combinatorial search to large distributed memory platforms

Learning Delaunay Triangulation using Self-attention and Domain Knowledge

Delaunay triangulation is a well-known geometric combinatorial optimization problem with various applications. Many algorithms can generate Delaunay triangulation given an input point set, but most are nontrivial algorithms requiring an understanding of geometry or the performance of additional geometric operations, such as the edge flip. Deep learning has been used to solve various combinatorial optimization problems; however, generating Delaunay triangulation based on deep learning remains a difficult problem, and very few research has been conducted due to its complexity. In this paper, we propose a novel deep-learning-based approach for learning Delaunay triangulation using a new attention mechanism based on self-attention and domain knowledge. The proposed model is designed such that the model efficiently learns point-to-point relationships using self-attention in the encoder. In the decoder, a new attention score function using domain knowledge is proposed to provide a high penalty when the geometric requirement is not satisfied. The strength of the proposed attention score function lies in its ability to extend its application to solving other combinatorial optimization problems involving geometry. When the proposed neural net model is well trained, it is simple and efficient because it automatically predicts the Delaunay triangulation for an input point set without requiring any additional geometric operations. We conduct experiments to demonstrate the effectiveness of the proposed model and conclude that it exhibits better performance compared with other deep-learning-based approaches

Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of G\"artner et al.\ applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange