Before revision:  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.
After revision:  Our first result shows that an algebraic abstraction of this problem is NP-hard . Assuming \mathrm{P}\neq\mathrm{NP} ) suggests that any polynomial time solution will need to exploit some geometric or topological intuition.