There are fundamental topological problems , such as unknot recognition and 3-sphere recognition, for which the existence of a polynomial time algorithm remains unknown. A major algorithmic tool behind some of the best known algorithms for these problems is normal surface theory. One of the key operations in normal surface theory is finding a non-trivial normal disc or sphere in a 3-dimensional triangulation. Our first result shows that an algebraic abstraction of this problem is NP-hard , which (assuming \mathrm{P}\neq\mathrm{NP} ) suggests that any polynomial time solution will need to exploit some geometric or topological intuition. Another key operation used by many topological problemsis to find a vertex normal surface of a certain type. We study two closely-related problems of this type: one we show is polynomial time, and the otherwe show is \mathrm{NP .
For many fundamental problems in computational topology , such as unknot recognition and 3-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms for these problems is normal surface theory. However, we currently have a poor understanding of the computational complexity of problems in normal surface theory : many such problems are still not known to have polynomial-time algorithms, yet proofs of \mathrm{NP finding a non-trivial normal sphere or disc in a 3-dimensional triangulation. We formulate an abstract problem that captures the algebraic and combinatorial aspects of this operation, and show that this abstract problem is NP-hard . Assuming \mathrm{P}\neq\mathrm{NP} , this result suggests that any polynomial-time procedure for finding a non-trivial normal sphere or disc will need to exploit some geometric or topological intuition. Another key operation , which applies to a much wider range of topological problems, involves finding a vertex normal surface of a certain type. We study two closely-related problems that can be solved using this operation. For one of these problems, we give a simple alternative solution that runs in polynomial time; for the other, we prove \mathrm{NP .
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of this type: one we show is polynomial time, and the otherwe show is \mathrm{NP
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