I have spend quite some time to come up with an algorithm to the solve Turnpike problem. turnpike algorithm in Julia. A causal look at says the complexity is beyond 2^n. But in practice it compute instanteously for a data set of about 16,000 delta numbers.
Here is the description of them problem. Suppose there is n increasing integers p_1 < p_2 , ... < p_n, we are given the difference of between every pair of points. The Turnpike reconstruction problem is to reconstruct the point set p_i from the distances of the form |x_i − x_j|.