Flattening fixed-angle chains is strongly NP-hard
Author(s)Demaine, Erik D.; Eisenstat, Sarah Charmian
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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.
12th International Symposium, WADS 2011, New York, NY, USA, August 15-17, 2011. Proceedings
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Algorithms and Data Structures
Springer Berlin / Heidelberg
Demaine, Erik D., and Sarah Eisenstat. “Flattening Fixed-Angle Chains Is Strongly NP-Hard.” Algorithms and Data Structures. Ed. Frank Dehne, John Iacono, & Jörg-Rüdiger Sack. LNCS Vol. 6844. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. 314–325.
Author's final manuscript