The traveling salesman problem with distances one and two
Mathematics of Operations Research
On the approximability of the traveling salesman problem (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
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The UPS Problem consists of the following: given a vertex set V, vertex probabilities (pv)v∈V, and distances l : V2 →R+ that satisfy the triangle inequality, find a Hamilton cycle such that the expected length of the shortcut that skips each vertex v with probability 1 - pv (independently of the others) is minimum. This problem appears in the following context. Drivers of delivery companies visit customers daily to deliver packages. For the company, the shorter the distance traversed, the better. For a driver, routes that change dramatically from one day to the other are inconvenient; it is better if one only has to shortcut a fixed route. The UPS problem, whose objective captures these two points of view, is at least as hard to approximate as theMetric TSP. Given that one of the vertices has probability one, we show that the performance ratio of a TSP tour for the UPS problem is 1=pmin, where pmin := minv∈V pv. We also show that this is tight. Consequently, Christofides' algorithm for the TSP has a performance ratio of 3=(2pmin) for the UPS problem and the approximation threshold for the UPS problem is at most 1=pmin times the one for the TSP.