Finding Tours through Near-Hamiltonian Paths? Say I have a connected graph.  I want to find a tour that visits each vertex at least once.  It's not always possible, though, for there to be a solution if there is a bridge in the graph.  Is there a term for this sort of problem or a recommended approach to solving it?  The motivation if it helps that I ultimately have in mind is traversing every station in a subway system.
There are no polynomial-time $\alpha$-approximation algorithms for TSP where $\alpha$ is a constant unless $\mathsf{P} = \mathsf{NP}$. 

However, for metric TSP there are approximation algorithms, e.g. Christofides algorithm. A simpler 2-approximation is obtained by taking an MST.