Question: Assume n cities are uniformly distributed in the unit disc. Consider the following heuristic for the n-city TSP. Let di be the distance from city

Assume n cities are uniformly distributed in the unit disc. Consider the following heuristic for the n-city TSP. Let di be the distance from city i to the depot. Order the points so that d1 ≤ d2 ≤ · · · ≤ dn. For each i 1, 2, . . . , n, draw a circle of radius di centered at the depot; call this circle i. Starting at the depot travel directly to city 1. From city 1 travel to circle 2 in a direction along the ray through city 1 and the depot. When circle 2 is reached, follow circle 2 in the direction (clockwise or counterclockwise) that results in a shorter route to city 2.

Repeat this same step until city n is reached; then return to the depot. Let ZH n be the length of this traveling salesman tour. What is the asymptotic rate of growth of ZH n ? Is this heuristic asymptotically optimal?

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