Numerical approximation of parametric oriented area-minimizing hypersurfaces
SIAM Journal on Scientific and Statistical Computing
Computing Least Area Hypersurfaces Spanning Arbitrary Boundaries
SIAM Journal on Scientific Computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
Minimal Surfaces Extend Shortest Path Segmentation Methods to 3D
IEEE Transactions on Pattern Analysis and Machine Intelligence
Optimal homologous cycles, total unimodularity, and linear programming
Proceedings of the forty-second ACM symposium on Theory of computing
The complexity of the normal surface solution space
Proceedings of the twenty-sixth annual symposium on Computational geometry
Hardness results for homology localization
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Maximal admissible faces and asymptotic bounds for the normal surface solution space
Journal of Combinatorial Theory Series A
Global minimum cuts in surface embedded graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
ACM Transactions on Mathematical Software (TOMS)
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Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. When the knot is embedded in a general 3-manifold, the problems of finding these surfaces were shown to be NP-complete and NP-hard respectively. However, there is evidence that the special case when the ambient manifold is R^3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1-dimensional subcomplex of a triangulation of the ambient 3-manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NP-complete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3-dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.