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For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case. 

Molecular programmers and nanostructure engineers use domain-level design to abstract away messy DNA/RNA sequence, chemical and geometric details. Such domain-level abstractions are enforced by sequence design principles and provide a key principle that allows scaling up of complex multistranded DNA/RNA programs and structures. Determining the most favoured secondary structure, or Minimum Free Energy (MFE), of a set of strands, is typically studied at the sequence level but has seen limited domain-level work. We analyse the computational complexity of MFE for multistranded systems in a simple setting were we allow only 1 or 2 domains per strand. On the one hand, with 2-domain strands, we find that the MFE decision problem is NP-complete, even without pseudoknots, and requires exponential time algorithms assuming SAT does. On the other hand, in the simplest case of 1-domain strands there are efficient MFE algorithms for various binding modes. However, even in this single-domain case, MFE is P-hard for promiscuous binding, where one domain may bind to multiple as experimentally used by Nikitin [Nat Chem., 2023], which in turn implies that strands consisting of a single domain efficiently implement arbitrary Boolean circuits. 

We give an overview of theoretical and practical aspects of finding a simple polygon of minimum ( Min-Area ) or maximum ( Max-Area ) possible area for a given set of n points in the plane. Both problems are known to be NP -hard and were the subject of the 2019 Computational Geometry Challenge, which presented the quest of finding good solutions to more than 200 instances, ranging from n = 10 all the way to n = 1, 000, 000. 

We introduce basic, but heretofore generally unexplored, problems in computational origami that are similar in style to classic problems from discrete and computational geometry. We consider the problems of folding each corner of a polygon P to a point p and folding each edge of a polygon P onto a line segment L that connects two boundary points of P and compute the number of edges of the polygon containing p or L limited by crease lines and boundary edges. 

Imagine t ≤ mn unit-square tiles in an m×n rectangular box that you can tilt to cause all tiles to slide maximally in one of the four orthogonal directions. Given two tiles of interest, is there a tilt sequence that brings them to adjacent squares? We give a linear-time algorithm for this problem, motivated by 2048 endgames. We also bound the number of reachable configurations, and design instances where all t tiles permute according to a cyclic permutation every four tilts.