Optimal algorithms for curves on surfaces

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can be continuously transformed to another, i.e., if they are homotopic. Our algorithm runs in O(n+k₁+k ₂) time and space, where closed curves C₁ and C ₂ of lengths k₁ and k₂, resp., on a genus g surface M (g≠2 if M orientable, and g≠3,4 if M is non-orientable) are presented as edge-vertex sequences in a triangulation T of size n of M. This also implies an optimal algorithm to decide if a closed curve on a surface can be continuously contracted to a point. Except for three low genus cases, our algorithm completes an investigation into the computational complexity of the two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. However, we make novel applications of methods from modern combinatorial group theory for an approach entirely different from previous ones, and much simpler to implement