The problem you describe is known as the Orienteering Problem, see for example this [survey][1], and also [this paper by Blum et al][2].


  [1]: http://www.inf.unibz.it/dis/teaching/SDB/papers/batch1a.pdf
  [2]: http://www.cs.cmu.edu/~avrim/Papers/orienteering-sicomp.pdf