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.
Research supported in part by NSF grant CDI-0941538.
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Demaine, E.D., Eisenstat, S. (2011). Flattening Fixed-Angle Chains Is Strongly NP-Hard. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_27
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DOI: https://doi.org/10.1007/978-3-642-22300-6_27
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