The problem of compatible representatives
SIAM Journal on Discrete Mathematics
Flat-State Connectivity of Linkages under Dihedral Motions
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Geometric and computational aspects of molecular reconfiguration
Geometric and computational aspects of molecular reconfiguration
Optimal binary space partitions in the plane
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
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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.