A general framework for searching on a line

Our main contribution is that the framework allows us to derive optimal competitive search strategies for variants of this problem that do not have a solution in the literature such as: (1) where the target is fixed and the searcher's cost at each step is α1x+β1 for moving distance x away from the origin and α2x+β2 for moving back with constants α1,α2,β1,β2, (2) where the target is moving and the searcher's cost at each step is a constant times the length of the step plus a fixed constant turn cost. Notice that the latter variant can have several interpretations depending on what the turn cost represents. For example, if the turn cost represents the amount of time for the searcher to turn, then this has an impact on the position of the moving target. On the other hand, the turn cost can represent the amount of fuel needed to make an instantaneous turn, thereby not affecting the target's position. Our framework addresses all of these variations.