**Reference**:- Erik D. Demaine, Hiro Ito, Jayson Lynch, and Ryuhei Uehara, “Computational Complexity of Flattening Fixed-Angle Orthogonal Chains”, in
*Proceedings of the 34th Canadian Conference on Computational Geometry (CCCG 2022)*, Toronto, Ontario, Canada, August 25–27, 2022, to appear. **Abstract**:- Planar/flat configurations of fixed-angle chains and trees are well studied in the context of polymer science, molecular biology, and puzzles. In this paper, we focus on a simple type of fixed-angle linkage: every edge has unit length (equilateral), and each joint has a fixed angle of 90° (orthogonal) or 180° (straight). When the linkage forms a path (open chain), it always has a planar configuration, namely the zig-zag which alternating the 90° angles between left and right turns. But when the linkage forms a cycle (closed chain), or is forced to lie in a box of fixed size, we prove that the flattening problem — deciding whether there is a planar noncrossing configuration — is strongly NP-complete. Back to open chains, we turn to the Hydrophobic–Hydrophilic (HP) model of protein folding, where each vertex is labeled H or P, and the goal is to find a folding that maximizes the number of H–H adjacencies. In the well-studied HP model, the joint angles are not fixed. We introduce and analyze the fixed-angle HP model, which is motivated by real-world proteins. We prove strong NP-completeness of finding a planar noncrossing configuration of a fixed-angle orthogonal equilateral open chain with the most H–H adjacencies, even if the chain has only two H vertices. (Effectively, this lets us force the chain to be closed.)
**Length**:- The paper is 7 pages.
**Availability**:- The paper is available in PDF (471k).
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Last updated March 28, 2023 by Erik Demaine.