Longest path in a graph with special property
As Raphael pointed out, this problem is also NP-complete. There is an easy reduction from longest path problem to this problem. Just take the graph of longest path problem and label all of the edges type-0. There won't be any type-1 edge. This labeled graph will have a longest path of length $k$ iff original graph has a longest path of length $k$.

This proves the NP-hardness of the problem. 

It is easy to see that the problem is in NP, the NDTM machine for the problem will be similar to the longest path problem. Both this add to make the problem NP-complete.

But if you are happy with any kind of approximation, then you can refer Approximation to longest path and modify the algorithm to suit your purpose. However be prepared that the modification may not be trivial.