**Reference**:- Erik D. Demaine and Sarah Eisenstat, “Flattening Fixed-Angle Chains Is Strongly NP-Hard”, in
*Proceedings of the 12th Algorithms and Data Structures Symposium (WADS 2011)*, Brooklyn, New York, August 15–17, 2011, pages 314–325. **Abstract**:- Planar configurations of fixed-angle chains and trees are well studied in polymer science and molecular biology. We prove that it is strongly NP-hard to decide whether a polygonal chain with fixed edge lengths and angles has a planar configuration without crossings. In particular, flattening is NP-hard when all the edge lengths are equal, whereas a previous (weak) NP-hardness proof used lengths that differ in size by an exponential factor. Our NP-hardness result also holds for (nonequilateral) chains with angles in the range [60° − ε, 180°], whereas flattening is known to be always possible (and hence polynomially solvable) for equilateral chains with angles in the range (60°, 150°) and for general chains with angles in the range [90°, 180°]. We also show that the flattening problem is strongly NP-hard for equilateral fixed-angle trees, even when every angle is either 90° or 180°. Finally, we show that strong NP-hardness carries over to the previously studied problems of computing the minimum or maximum span (distance between endpoints) among non-crossing planar configurations.
**Length**:- The paper is 12 pages.
**Availability**:- The paper is available in PDF (239k).
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Last updated September 20, 2023 by Erik Demaine.