Combinatorial computation of characteristic classes

Consider a triangulated smooth manifold. Characteristic classes are invariants of fiber bundles that are some special base cohomology elements. They can be used to partially classify fiber bundles and manifolds. Then a naturally arising question is the following: is there a “reasonable” algorithm to compute characteristic classes of the manifold tangent bundle using exclusively the combinatorial data arising from the triangulation? The aim of the talk is to present some known answers and an arising application, as well as discuss further problems in the area.