Universal Approximations in Network Design — 17 June , 2011

The standard courier does the following: every day he receives a fresh list of addresses, from which he computes a TSP tour (only using the addresses he received that day). The lazy courier, being lazy, would like to avoid the work of having to compute a new tour every day. He decides to use all of the addresses in the database to compute a single TSP tour. He will then follow that tour every day, stopping only at the addresses in that day’s assignment, and shortcutting wherever possible. For instance, if the single TSP tour computed over the set {A,B,C,D,E, F,G,H} of addresses is A → B → C → D → E → F → G → H → A, and the list of addresses to visit on a given day is {A,D,E,G}, the lazy courier will travel along the route A → D → E → G. Clearly, the computation done by the lazy courier each day is minimal, while the standard courier needs to solve a (possibly new) TSP problem every day. On the other hand, it is not hard to see that on some days the lazy courier will travel more than the standard courier for a given address list. This raises the following question: how good of an approximation is this universal tour (computed by the lazy courier) as compared to the method of computing a separate tour for each day’s delivery assignment?